WASHINGTON—The number of Americans applying for unemployment benefits fell to the lowest level since 1969, the latest sign the labor market is firming after years of steady job growth.

Initial jobless claims, a measure of layoffs across the U.S., fell 24,000 to a seasonally adjusted 209,000 in the week through April 21, the Labor Department said Thursday. That was the lowest level since Dec. 6, 1969, early in the Nixon administration.

Economists...