Stanley used precision iron play and deep drives to match a tournament record with a 54-hole total of 18-under 198, equaling a mark set by Woody Blackburn in 1985 and tied by Tiger Woods in 2008.
Should Stanley hang on at Torrey Pines in California, he will record his first PGA Tour victory.
Tied for second at 13-under 203 after three rounds were John Huh and John Rollins, both of whom shot 68s.
Sang-Moon Bae (72) and Bill Haas (70) shared fourth place one shot further back at minus-12.
Celebrity Families of 2014 [PHOTOS]