This question has been asked and answered many times before, but the short answer is the late 19th to early 20th century, with first a general awareness of America's latent power through its industrial might (America surpassed Britain as the largest economy in the world in the 1870s; European ministers were upgraded to the rank of ambassador in 1892), followed by increasingly assertive US foreign policy actions:

• In 1895, Britain gave in to US pressure to settle a border dispute between their colony in Guyana and Venezuela through arbitration.

• In 1898, the US rapidly defeated Spain and acquired an overseas colonial empire in the Caribbean and Pacific, with Cuba and the Philippines coming under American domain. Hawaii was also annexed that same year.

• In 1900, the US sent 6,300 troops as part of a multinational expedition to crush the Boxer Rebellion in China. The US then submitted a note to the European powers (the Open Door policy) that said it expected its economic and political interests in China to be respected, laying down the marker for an independent US policy in the Far East.

• In 1901, Britain formally conceded to the US the right to build and operate a canal in Panama.

• In 1905, the US brokered a peace settlement that ended the Russo-Japanese war, which bagged Theodore Roosevelt the Nobel Peace Prize.

It was also around this time that the US's industrial might was being converted into tangible military power. The US navy grew threefold during Theodore Roosevelt's presidency, from 11 battleships to 36. In fact, the US had the 3rd largest navy in the world, after Britain and Germany, by the start of WWI. Europeans only begrudgingly conceded these changes, with the British prime minister Lord Salisbury lamenting in 1902 that, "It is very sad, but I am afraid America is bound to forge ahead and nothing can restore the equality between us. If we had interfered in the Confederate War it was then possible for us to reduce the power of the United States to manageable proportions."

So there was first a recognition of US potential, followed by increasing American assertiveness in the world, and then finally European acquiescence and accommodation to US interests and power by the early 20th century.

Sources:

From Colony to Superpower by George Herring.

The War that Ended Peace: The Road to 1914 by Margaret MacMillan.

There's some debate over this, although historians typically choose one of three dates: 1898, 1919, or 1945.

I hew toward the belief that the US was seen as a "Powerful Nation" after WWI, but the US didn't become a truly dominant power until the end of WWII. I'll make that argument in the following paragraphs by summarizing many books by various historians, but keep in mind that this argument is simply my interpretation and not the unanimous conclusion of all historians.

From the Early Republic and through the Civil War, the United States was primarily a regional power, albeit one that was strongly integrated into global networks of trade, politics, and cultural transmission. In the decades immediately following the American Revolution, the United States more often than not depended on the acquiescence or acceptance of other global powers to conduct its foreign affairs. In these years the United States was more concerned with attaining its rightful place among the community of sovereign nations than it was asserting its international power over other states.

As David Armitage has shown in his Declaration of Independence: A Global History, the Declaration was America’s first state-making document and was written in the hopes that other states would support its arguments and ideas. While historians have typically focused on the Declaration as a human rights document, Armitage rightly points out that it was primarily a document claiming sovereignty if not over a specific territory at least over a specific people. It established who Americans were, where they lived, and who did (and did not) have the right to govern over them. But perhaps most importantly, the document stated that Americans held the right to be free from the dictates or control of a foreign power; in other words, Americans declared not just the overthrow of British imperial rule but also the establishment of a new government led by Americans, and one that the world should see as a legitimate state within the international order.

Americans during and immediately after the Revolution backed up this idea with action. As Eliga Gould argues, they attempted to demonstrate their sovereignty by acting like other sovereign powers, specifically by gaining dominion over foreign peoples. It is for this reason—international recognition of American sovereignty—that the United States began expanding into Native American territory and declared the Monroe Doctrine so early in the country’s history. The American desire for international legitimacy and respect for U.S. territorial sovereignty over North America seemed to be accepted with the American purchase of France’s Louisiana Territory. Ceding and taking territory was something only a legitimate state could do. However, America's global power was simultaneously questioned in a several ways: for one, the British impeded American expansion into Canada during the War of 1812 while simultaneously making a mockery of American sovereignty through the impressment of its sailors and the burning of Washington, D.C.

Americans in the early republic did, however, exert a fair deal of global cultural power. Seth Cotlar has shown how American and French radicals traded ideas about popular politics, cosmopolitanism, and egalitarian forms of economics through a transnational network of newspaper editors, printers, and booksellers. In this way, American and French radicals crafted and disseminated a potent form of revolutionary thought, at least until the Terror of the French Revolution delegitimized radicalism in the new United States. American sailors also played an integral role in connecting the United States to the broader world, as Brian Roleau shows in With Sails Whitening Every Sea. Sailors acted as key transmitters of American culture throughout the port cities in which they put harbor. These sailors traded extensively with foreign peoples, thereby placing American money and goods squarely into the global economic market. Despite these early instances of global cultural interaction and economic integration, the United States still did not hold the international power or prestige to be labeled a powerful nation or even a rising power. Throughout the Early Republic era, the United States was at best the primary regional power in the Western Hemisphere.

During the Civil War era, too, U.S political leaders were often left making decisions based on how they thought foreign powers would react. As Howard Jones notes in his history of Union and Confederate foreign policy, Blue and Grey Diplomacy, fears of British and French intervention on the Confederates’ side played a major role in determining many of the Union’s actions. Only after the Emancipation Proclamation turned the war into a moral war over slavery, and only after the Union victory at Antietam demonstrated that the Union held the upper hand, did the threat of British intervention finally subside.

Despite the diminished status of American's global military and political power during the Civil War era, the conflict also became a cause célebrè for many European nationalists. As Don Doyle shows in The Cause of All Nations, many of these nationalists saw the American Civil War as a fight in favor of liberalism, and they urged Lincoln to state unequivocally that the Union was fighting to rid the United States of slavery. In this sense, then, the United States might have lacked military or political power during the Civil War, but it retained a sizeable degree of cultural influence in the world as Europe’s liberals rallied under the Union cause.

It was, however, in the late 19th century that the United States began to exert its hard power abroad, although it did so only intermittently. The War of 1898 demonstrated that the United States could establish itself in a role similar to that of other major world powers, namely as a state that could rule over foreign peoples in a far away territory. Beyond the War of 1898, the United States was expanding its global reach in other ways as well. As Katherine Unterman has argued in Uncle Sam’s Policemen, it was during this period that the United States began encroaching on the sovereignty of other countries in pursuit of American fugitives. Often faced with cumbersome extradition laws, American officials simply decided to ignore those laws--agreed to by international treaty-- and came into the foreign countries, captured the fugitives, and returned them to the United States for trial. Such actions demonstrated that the United States had become more assertive in its foreign policies by showing a disregard toward the sovereignty of other states. Many other historians have discussed how US cultural and economic influence in the world dramatically increased at this time, but I'll skip over that topic.

By the end of the 19th century, the United States appeared to have both the ability and the will to exert its influence as a truly global power; however, in the aftermath of World War I, the United States turned inward. As Adam Tooze has argued in The Deluge and Kiran Klaus Patel has argued in The New Deal: A Global History, the United States held a unique role in the interwar era, one in which it had the ability to save the global economy and stop the worldwide slide toward economic autarky. Yet, beginning with the failure to approve the Treaty of Versailles, the United States stepped back from the global stage beginning in 1920. President Wilson’s initial call for national self-determination, which inspired nationalist movements throughout many European colonies, was cast aside by Wilson himself in favor of maintaining the world’s status quo of European imperial domination. Moreover, throughout the 1920s-30s the United States repeatedly failed to accept global leadership by withdrawing from several currency stabilization plans and scuttling international economic conferences. In other words, throughout the 1920s and until American involvement in World War II, the United States might have had the ability to exert its influence as a truly global power, but it did not carry the will to do so.

This changed, however, during World War II. The American experience in depression and war provided the foundation for the United States' postwar status as a genuine world power. Presidents Roosevelt and Truman believed that democracy was under threat from Nazism abroad and radicalism at home; consequently, they thought it was necessary for the state's survival to protect the American people by lifting them out of poverty and, soon thereafter, defeating the Nazi menace. These ideas carried over into the postwar era, although the enemy shifted from the Nazis to Soviet communism. Emerging out of World War II as one of two superpowers, the United States clearly had the economic, military, and political clout to act as a global power; all other states had been devastated by the war and their first priority turned inward toward rebuilding and recovery. The United States, relatively unscathed from the war, became the torch-bearer for democracy and the financier of Western European reconstruction. At the same time, the American people and their political leaders believed it was necessary to expand democratic capitalism's reach over the entire world. Failing to do so would allow communism to expand and, in time, threaten the very existence of the American way of life. In this way, the post-WWII period represented a unique moment for the United States. It was a time in which American military, economic, and political power was predominant, and that dominance melded with a staunch commitment to protect democratic capitalism throughout the world.