The most common answer is when they vanquished us (Spain) in the Spanish-American war (1898-1899). That war was universally seen as the end of an empire (the Spanish) and the beginning of another (the american) with the annexation of the Philippines, Puerto Rico, Guam and the near-annexation of Cuba.

The US didn't really become a superpower overnight. But there were moments at which we can see it's rise as a power.

The first is probably the Venezuelan crises, 1895 and 1902. The first was a watershed moment. Venezuela claimed half of British Guyana as it's territory. The US demanded that it arbitrate in the matter as it fell under the US sphere of influence according to the Monroe Doctrine. The British eventually acquiesced to this demand the the US judged the matter to the satisfaction of the British. However, the implication was in fact the UK accepted the Monroe doctrine, at least militarily, not economically.

A second is probably the end of the first world war. At which point the US overtook the UK as the world's leading nation in terms of GDP and became a centre of finance as well as industry. However although the US played a large part in the treaty of Versailles, it prefer to remain isolated from European affairs and did not really impose itself.

The last is probably the end of the second world war, at which point a nuclear armed United State's economy makes up over half of the World's GDP and is truly in the position of a superpower; it can economically and militarily enter into a war anywhere in the world, with any nation and stand a good chance of winning. With the creation of NATO and the UN, the US is firmly involved in the affairs of the world and it's foreign policies become integral to the nation.