Stenson, one of the best players not to have won a major championship, rolled in a 40-foot putt to save par at the final hole and join Americans Dustin Johnson and Russell Henley in a tie for third at the windy Golf Club of Houston.
American journeymen Jim Herman and Jamie Lovemark carded 67 and 70 respectively to head a crowded leaderboard at 11-under 205.
Stenson, winless on the PGA Tour since the 2013 Tour Championship, fired seven birdies and one bogey, but needed to get up-and-down to save par on the final three holes.
“That really keeps us in contention,” he told PGA Tour Radio. “It was tough coming in. It was gusty conditions.”
Stenson has been more successful recently on the European Tour, winning the season-ending World Tour Championship in Dubai in 2013 and 2014.
A win on Sunday would provide a timely confidence boost heading to next week's Masters, where the world number seven will try to break his major drought.
“I'm going to try once again to play a good final round and we'll see where that takes us,” Stenson, 39, said.
“It's a really bunched up leaderboard and I'm expecting the lead to change hands more than once tomorrow.”
A fast start from Lovemark and a sturdy finish from Herman pushed the pair of Americans into a share of the lead. Herman birdied two of his final seven holes for his bogey-free round, while Lovemark made three straight birdies beginning at the second hole.
Neither the 38-year-old Herman nor Lovemark, 28, has won on the PGA Tour. A victory on Sunday would earn a Masters invitation.
“There is a lot to play for tomorrow, but I'm not really worried about that,” Herman told Golf Channel.
“I'm going to keep doing what I have been doing the last three days and that's keeping the ball on the fairway and out of the hazards, and giving myself as many birdie chances as I can.”
Lovemark charged into the lead at 12-under with his hot start. But he played his final 14 holes in one-over.
Overnight leader Charley Hoffman struggled with a 74 to slip three shots behind, while Masters champion Jordan Spieth (70) is five back. – Reuters
Original source: Stenson vaults within shot of lead