Wage growth is breaking out in an unexpected corner of the U.S. economy: the nation’s restaurants and bars.

Food-service employment has surged since the recession ended nearly six years ago, growing twice as fast as overall payrolls. But those gains had largely failed to translate into better wages in the sector, until recently. Restaurant wages zoomed up to an annualized pace of more than 3% in the second half of last year from below a 1.5% pace in the first half of 2013, according to the Labor Department. Private-sector...