Rodgers, 30, is chasing a first tour title in his 235th career start, one which would gain him a spot in the field for the Masters, the first major championship of the year at Augusta National next week.
"It's going to require good golf," Rodgers said. "There's a lot of guys chasing and great players out here. Obviously there's no better place to be than to have an advantage, but it's going to take a great round in order to get it done.
"I think we all know what's at stake with a win out here," he added. "I haven't gotten the job done in my career, but it's quite a thrill. This is why I play, why I compete and I can't wait to get out there tomorrow."
Rodgers's 12-under-par total of 204 at TPC San Antonio left him one stroke in front of 2019 champion Conners, who carded a 69, with veteran Matt Kuchar alone in third after a 69 for 207.
Overnight leader Rodgers led Conners by two after his third birdie of the day at the 17th, where he rattled in a nine-foot putt.
But his tee shot at 18 settled under a tree right of the fairway.
Forced to punch out, he found a bunker with his third shot and couldn't get his par-saving 13-foot putt to drop.
Canada's Conners, who got up and down for birdie from a greenside bunker at 17, had a chance to grab a share of the lead but missed a seven-foot birdie putt at the last.
"Didn't get off to the best start," said Conners, who had seven birdies, a double bogey and two bogeys in his three-under effort. "Made a few mistakes on the first nine, but really happy battling back and making some birdies on the back nine today. Lots of positives and good momentum from that nine holes heading into tomorrow."
Kuchar, chasing a 10th PGA Tour title and his first since the Sony Open in 2019, applied some pressure with six birdies - three each side of a mid-round bogey.
A birdie at 17, where his 35-foot eagle putt burned the edge of the cup, moved him within one of the lead.
But his tee shot at the par-five 18th went left into a mounded cactus, costing him an unplayable lie on the way to a double-bogey.
"This was definitely my best day ball-striking by a long shot, was playing some good golf," Kuchar said but added: "It is a tough, demanding golf course and I paid the price with a poor tee shot on the last. But all in all, listen, it was a good day of golf out there."
