Ah, this is a great question!
Economic sanctions have existed for thousands of years. The earliest known economic sanction that I'm aware of occurred in the 5th century BCE when Pericles, an Athenian statesman, ordered a trade embargo against Megra, a city-state that had allied with Sparta.
From that point in history until the 20th century, economic sanctions were infrequent and generally unilateral (i.e., one state imposing sanctions on another state), and. As the notion of the nation-state as a sovereign political entity developed in Europe during the Enlightenment and through the 19th century, philosophers and politicians began to seriously conceptualize a world in which international disputes could be solved via multilateral diplomacy and soft coercion instead of violence and war.
Immanuel Kant outlined such a world in his essay Perpetual Peace, where he describes the conditions that need to be agreed upon by the world's leaders in order to prevent war.
As the realities set in of just how brutal and nightmarish modern war would become in the aftermath of World War I, many American and European leaders turned to Kant for the philosophical inspiration that lead to the creation of the ill-fated League of Nations. The League of Nations was the first major international governmental organization in the world to have the state goal of preventing war by using economic sanctions to thwart aggressor nations. As we all know, the League of Nations failed miserably.
After Europe did the wash, rinse, repeat thing with WWII, a new ideological movement in international relations emerged. European leaders and thinkers realized that economic sanctions failed with the League of Nations because a) the LN had no way to really enforce sanctions, b) the economic sanctions it could impose only served to drive Germany back onto the war path, and c) there needed to be a stick and carrot approach that incentivized cooperation between competing nations. This ideology, sometimes called "internationalism" or "supranationalism" in different contexts lead to the creation of the United Nations and the predecessor organizations to the European Union.
Let's back up for a moment, though. We take for granted that today, Germany and France share one of the closest diplomatic relationships in the world. After the war ended in 1945, many in war torn France wanted Germany to suffer, while many still had the memories fresh in their mind from what happened the last time the international community punished Germany too severely. At any rate, everyone in France was still very distrustful of Germany, for good reason.
If I could point to one moment in history that was so monumental for defining our world today that most people might not know about, it would be the creation of the European Coal and Steel Community in 1951, the first precursor to the European Union. Signed by Belgium, France, West Germany, Italy, the Netherlands and Luxembourg, what the ECSC did was pool the nations' coal and steel production on a common market in order to prevent trade wars over their natural resources, which encouraged significant growth and development in the war torn nations' industrial sectors.
BUT more importantly, the ECSC was designed to encourage transparency between nations when it came to what they were producing with coal and steel - the primary building blocks necessary to fuel the engine of war. Because France could see what Germany was doing with its coal and steel, fears of a reemerging German threat subsided rather quickly.
So that is the "carrot" approach that made the "sticks" carry more weight. In order for economic sanctions to be effective, they have to make economic cooperation look attractive.
Which brings us to the creation of the United Nations. You know how drug dealers will often give a new customer their first hit for free because they know they'll come back for more? Ok, terrible analogy, but the world leaders at the time had to make the United Nations an attractive organization for other nations to participate in. Under the umbrella of the UN are the World Bank, the International Monetary Fund, and the UN Development Programme. In essence, these three organizations collectively provide economic assistance in the form of low interest loans, grants, aid, debt restructuring while promoting trade and economic development.
So if the LN's approach to economic sanctions was: "If you don't play nice, you don't get to play with anyone" then the UN's approach is "If you don't play nice, not only do you not get to play with anyone, but you won't get any of our free-ish money."
The thought is that aggressive nations are generally less democratic, poorer, and are going to be more dependent on foreign aid. If the UN pulls a nation's aid away, then the leaders of that nation will have to deal with an angry populace. This has had mixed results - the economic sanctions imposed on Iraq during the 1990s caused countless deaths, and many people argue that economic sanctions only end up punishing the people who are the worst off to begin with.
Many will point to the UN's many failures to solve international crises via economic sanctions or "peacekeeping" as proof that the organization is just the League of Nations 2.0. I can't say there isn't merit to that argument, but I do think it's important to consider that in the scope of history, the UN is still in its infantile stage.
I won't bother delving into the whole argument over how effective economic sanctions are, but I do want to briefly talk about how all of this can help us understand the relationship between the United States, Europe, and the rest of the world.
The divergent ideology that arose in response to internationalism in the 1950s is interventionism, or neoconservatism (which has a bit of negative connotation now). Neoconservatism has its roots in the Cold War policy of containing communism through military intervention, but it also sprung about from the United States' often tepid relationship with the UN.
It all comes back to Kant. Whereas internationalists envision perpetual peace, interventionists take the Hobbesian view that the world is in a state of chaos, and that nations must be able to act unilaterally to secure their own destiny. This is really the difference in how Americans and Europeans generally see the world. This isn't to say that neoconservatives see no value in economic sanctions, but rather that they would say internationalists are far too idealistic. Similarly, internationalists will accuse interventionists of being too trigger happy and pessimistic. And it all comes down to how the two camps view economic sanctions as a means to ensure peace.
That debate rages on.
Sources:
Thucydides, History of the Peloponnesian War.
Immanuel Kant, Perpetual Peace: A Philosophical Sketch.
John McKormick, Understanding the European Union.
Robert A. Pape, Why Economic Sanctions Do Not Work. International Security. Fall 1997, pp. 90-136.
Robert Kagan, Power and Weakness. Policy Review, June 2002, pp. 3-28.
Edit: Some minor changes for clarity.