I am presuming you are referring to international influence outside of the North American sphere.

The best answer for the political part would probably be the Spanish-American War. During that conflict, US forces won decisive simultaneous victories against a European (Spanish) military in two hemispheres, humiliating an "Old World" power and stripping it of the last of its colonial possessions. This surprised much of the world as it showed the US' ability to project decisive naval and ground power on a global scale against a "modern" fighting force.

The economic answer is much more nebulous but as far as the groundwork for today's world economy, which is driven in very large part by the US (although this has been changing in past couple of decades), the best answer would probably be the aftermath of WW2. In the aftermath of the conflict, the US was the only major industrialized nation not decimated by the war. This enabled the US to rebuild Western Europe through the Marshall Plan. The non-communist global economy soon became effectively realigned around the American (and, to a lesser extent, British) economy, with even European initiatives like the Coal and Steel Community (which later evolved into the EU) being deeply codependent on the US economy, not to mention Japan and later South Korea being rebuilt as bookies economies tightly linked with US markets. WW2 also had the effect of boosting the US' industrial base without suffering bombing and invasion of its cities, while millions of servicemen who returned home were able to attend college and secure employment amidst a booming US economy and rapidly expanding population that lasted for decades.

Edit: spelling