WASHINGTON—Unemployment in September hit the lowest level since the Vietnam War, with little indication it is going to shoot back up in the near term.

The jobless rate fell to 3.7%, the lowest since December 1969, the Labor Department said Friday. Employers added 134,000 jobs to payrolls, a record 96th straight month of gains. Wages rose 2.8% from a year earlier, a solid if still unspectacular rise.

“This...