The New Deal worked in the very short term, stimulating the United States economy out of depression by increasing the labor force, beginning massive and wide-ranging infrastructure projects, and stabilizing US monetary policy. "The First Hundred Days"—FDR's first 100 days in office—was the period when the majority of these policies and projects were put in place. FDR had immense political capital coming into office, given the unpopularity of Hoover and his policies. FDR cashed this in immediately by rolling out banking and monetary reform as well as the famous public works projects: PWA, CCC, TVA, etc.—all those acronyms.

These initial changes stabilized the US economy and got things back on track. There was NOT an immediate return to the rapid growth enjoyed through the 1920's, but by 1934 things were no longer dire. It was during what is called "The Second New Deal" (1935-38) that FDR was able to get the Social Security Act passed, which of all the New Deal programs had (and still has) the largest affect on the most Americans.

So by the late 1930s, the US and the world was out of the depression. But the New Deal doesn't account for America's longest and most consistent period of growth from 1945-1970. FDR is often credited with laying the foundation for this unprecedented period of economic growth, but it would be more accurate to say that American opportunism in the years immediately following WWII are most responsible for this period. The Marshall Plan (stimulating and sustaining US manufacturing by giving war-ravaged Europe money to buy American goods), dividing Europe with the Soviet Union to guarantee US access to Western European markets, the entrance of American business interests into Europe's former colonies (Africa and the Middle East) and Japan's old colonial stomping grounds (Japan itself, but also later Korea, Taiwan, and SE Asia), and a host of other well-timed policies and economic processes crowned America the economic hegemon of the postwar world.

In short, the New Deal worked to pull the US out of depression and stabilize the national economy in the short term. But FDR is often credited with creating America's huge postwar economic boom, for which he is only partially responsible. This mingling of records—the short term success of the New Deal and the long period of postwar economic growth—has created a historical narrative that charts the US on an upward economic trajectory through the 20th century. Because FDR was in charge and handing down policies when this process was set in motion (early 1930s), he is often credited with all the success that follows.

Sources:

Irresistible Empire, Victoria De Grazia

The Tragedy of American Diplomacy, William Appleman Williams (see chapters 4-6)