The word "super power" has direct connotations with post-WWII diplomacy between the two "super powers" of the United State and the Soviet Union. A better term might be "world power" or, at least, a major player in the international diplomatic game.
To answer your question, the Spanish American War in 1898 can be seen as America's first attempt to establish a sort of overseas colonialism that had already been practiced (and was, of course, still being practiced) by a number of European powers (and Japan, for that matter). The acquisition of Puerto Rico and the Philippines from the Spanish, along with the de facto exercise of power over Cuba through unequal treaties, meant that the United States now stood as a culturally-European nation with overseas territories with large indigenous populations to manage.
The westward expansion of the United States on the American continent throughout the previous century should also be considered a kind of colonialism/imperialism, but the consequences of the Spanish-American War were fundamentally different because the maintenance of these new territories required the use of well-developed naval power and globe-spanning economic influence to maintain control. The acquisition of the Philippines, for example, gave the United States a voice to support "open door" trade policies with China in lieu of letting the European powers carve up broader spheres of exclusive influence and control. Without direct territorial control of land in the geographic vicinity of the "flashpoint" area - in this case, China - the United States would not have had a seat at the table.
Indeed, the presence of American soldiers in the effort to contain the Boxer Rebellion in China, and the diplomatic aftermath that also witnessed American power at the table, can be seen as symptomatic of this new-found position as a "world power" in the colonial sense after the Spanish-American War. The same can be said of Teddy Roosevelt's brokerage of the peace treaty between Russia and Japan in 1905 after the aptly-named Russo-Japanese War due to the diplomatic weight that America had begun to acquire at this time. American economic influence was also growing throughout Central and South America at this time as well. American firms owned large portions of the arable land and the industries of many nations in Latin America, meaning that America exercised a sort of pseudo-imperial power over these nations without having to exercise direct colonial control as a European nation would do in, for example, sub-Saharan Africa.
Edit: My personal favorite text on this subject comes from Walter LaFeber's The American Age: United States Foreign Policy at Home and Abroad, 1750 to the Present. The second volume starts with the Spanish-American War and the following years as America began to perform its role as a "great power"(as /u/skgoa puts it).
I believe the US became the #1 Economic power (although it was a very close 1 2 for a while) approximately in the 1910-1915 range, just before WWI. However they also become the #1 Military power during the course of WWII, so I would say their Super Power status has been cemented since about 1944-1945. They were "relevant" since about the late 1880s however, and around the turn of the 20th Century, they had gained enough steam from initial immigration and the Industrial boom, that they would have been approximately on par with most of the large European powers.
The US has actually been the largest economy since 1871, contrary to what someone else commented. It was really recognized as a "great power" in 1898 after the Spanish-American War. This status was solidified in 1917 by US entry into World War I, and by the Paris Peace Conference in 1919 the US was first among the great powers. But it backed away from playing such a role during the 1920s and 1930s and reduced its military power significantly. The US entry into World War II in 1941 and rapid militarization returned it to active military involvement in Europe and Asia. By the end of the war in 1945, the US was recognized as a global superpower. With the fall of the Soviet Union and communist bloc from 1989-1991, the US become the sole superpower.