If you mean "globally prestigious," the United States was considered a "second-tier" educational/scientific state in the early 20th century. The "first-tier" were the big three in Europe (United Kingdom, France, Germany) who dominated in scientific publications, discoveries, and educational institutions. If you were a superstar student in the United States in the early-20th century, you usually got your PhD abroad. (The other main "second-tier" states at that time were Russia and Japan, as well as a smattering of other European countries.)

This was beginning to change during the interwar period, in part because those students who got PhDs abroad came home and began educating a new generation of American academics. By the 1920s you didn't have to go abroad to get a great education; you could do it in your own back yard. Some of this involved major changes to how American universities worked, as well; in the 1930s, Harvard University, under James Bryant Conant, adopted a "German model" in its approach to research, for example, with a higher emphasis on research output as the criteria for promotion and tenure.

But the big acceleration took place during and after World War II. This was caused both by the fact that the "big three" countries were so wounded and disrupted in that war, and by the fact that there was, from Germany in particular, an exodus of academics because of the oppression of the Nazis. Some of these were famously Jewish academics, who were fired from their university jobs in one of the first laws passed by the Nazis in 1933 (universities in Germany are part of the civil service, and the Nazis banned all people with Jewish ancestry from working in the civil service). But there were others who fled as well.

The United States emerged from the war with many top-flight European experts as part of its university systems, with a new drive for educational funding for scientific and defensive purposes, and with a massive population of GIs looking for education under the GI bill. All of these acted like rocket fuel for the prestige and power of US universities; the Cold War was by and large very good for academic institutions, even if it could sometimes be quite stifling to individual academics.

In terms of your chicken and egg question — there was a mutual development. Had US science and technology not been in a strong position at the beginning of World War II, it would have been difficult for the nation to become a superpower in the first place. But the war had a transformative effect on the entire country, including universities.

There are several good histories of the rise of the Cold War university; Stuart Leslie's The Cold War and American Science is a very good place to start.