The American produced his third straight bogey-free round and sunk eagles at the par-five eighth and 13th holes of the PGA West Nicklaus course for a round of 63 to but him 22 under overall.
That put him five ahead of a group of five players, led by joint overnight leader Roberto Castro, who was three under on the La Quinta course.
Alongside him were former Open champion Stewart Cink, Charles Howell, Charley Hoffman and John Rollins, who produced an eight-under round on Nicklaus.
"I've been aggressive," Stallings told www.pgatour.com. "I've been smart when I got out of position. Just because I played really good, doesn't mean that I've hit it perfect on every single hole. I scrambled a little bit, but when I got out of position off the tee, I played smart.
"I hit it in the middle of the green and just kind of accepted par and moved on and was really aggressive when I had the opportunity."
The performance of the day came from another American, Brian Struard, who was 10 under on Nicklaus.
His round included two eagles, eight birdies and two bogeys. At one point he was on for a rare round of 59, but he went par-bogey at the 17th and 18th and ended up in a group of eight players on 16 under.
Sweden's David Lingmerth was leading the European challenge on 15 under, while Britons Martin Laird, Greg Owen and Ross Fisher, on 12, 11 and 10 under respectively, made it through to the final day.
With the cut being after day three so all players have played each of the three courses, the mark was set at 10 under, with only six of the 156-strong field over par.