Stallings' performance was his third straight round without a bogey at La Quinta Country Club and enabled him to end the day at 22-under-par 194 as he seeks his third career PGA Tour victory.
Second-round co-leader Roberto Castro registered a 3-under 69, dropping into a second-place tie at minus-17 with Stewart Cink (66), Charles Howell III (67), John Rollins (64) and Charley Hoffman (67).
Meanwhile, Phil Mickelson overcame flu-like symptoms to make the cut in his 2013 season debut.
The left-hander finished Saturday at 11-under after carding a round of 6-under 66.
Notable deaths of 2014 [PHOTOS]