WASHINGTON—A robust economy drove U.S. wages higher in August, new evidence that workers are gaining bargaining power with their employers as the nation’s pool of available labor tightens.

The Labor Department reported Friday that worker payrolls expanded by 201,000 in August and private-sector hourly wages grew 2.9% from a year earlier. The payroll gain marked the record 95th consecutive month of job growth—much of it near or slightly below that 200,000 monthly mark—and the year-over-year wage increase was the largest since...