He never really brought America ought of depression until WW2 occurred, though. As far as recovery goes, his was slower than most other countries in the world at that time.
His economic policies could be credited with saving America from the worst (although America's recovery was by no means exceptional compared to, well, anybody else), but the economic boom occurred because of WW2 armament.
https://en.wikipedia.org/wiki/New_Deal#World_War_II_and_full_employment
This is one of the takes from that Wiki article:
According to Christina Romer, the money supply growth caused by huge international gold inflows was a crucial source of the recovery of the United States economy, and that the economy showed little sign of self-correction. The gold inflows were partly due to devaluation of the U.S. dollar and partly due to deterioration of the political situation in Europe.[85] In their book, A Monetary History of the United States, Milton Friedman and Anna J. Schwartz also attributed the recovery to monetary factors, and contended that it was much slowed by poor management of money by the Federal Reserve System. Former Chairman of the Federal Reserve Ben Bernanke agreed that monetary factors played important roles both in the worldwide economic decline and eventual recovery.[86] Bernanke also saw a strong role for institutional factors, particularly the rebuilding and restructuring of the financial system,[87] and pointed out that the Depression should be examined in an international perspective.[88]
My calculations suggest that in the absence of these stimuli the economy would have remained depressed far longer and far more deeply than it actually did. This in turn suggests that any self-correcting response of the U.S. economy to low output was weak or nonexistent in the 1930s.
https://web.archive.org/web/20130117093624/http://elsa.berkeley.edu/~cromer/What Ended the Great Depression.pdf
She has also researched the causes of the Great Depression in the United States and how the US recovered from the depression. Her work showed that the Great Depression occurred more severely in the US than in Europe, and had somewhat different causes than the Great Depression in Europe. Romer showed that New Deal fiscal policy measures, though innovative, were very insufficient, and dwarfed by Hoover's tax increase two years earlier.[8] However, accidental monetary policy played a large role in the US recovery from depression. This monetary policy came first from the devaluation of the dollar in terms of gold in 1933–1934, and later from the flight of European capital to the relatively stable US as war in Europe became more likely.[9]
https://en.wikipedia.org/wiki/Christina_Romer
It seems to be a bit of a myth that the New Deal caused a Golden Age to occur for the United States.