A surge in U.S. incomes last year delivered the first significant raise for the typical family after seven years of stagnant and declining earnings, the result of sustained job growth finally lifting a broad swath of American households.

The median household income—the level at which half are above and half are below—rose 5.2%, or $2,798, to $56,516, from a year earlier, after adjusting for inflation, the Census Bureau said Tuesday.

The...