Varner was three shots in front of Wesley Bryan and Brian Harman among the early starters at the Sedgefield Country Club.
Varner had eight birdies, including four in a row midway through the round, without a birdie his first 15 holes. He finished with three pars.
Bryan shot a 65 to match his season's best round, set last month in Detroit.
Former U.S. Open champ and past Wyndham winner Webb Simpson was among a large group at 66.
A pair of major champions who challenged at the PGA Championship last week struggled in the opening round.
Brooks Koepka had a 72 and Justin Rose a 73.