Dunlap, a 20-year-old sophomore at the University of Alabama, posted 10 birdies and an eagle - on the par-5 sixth - at La Quinta Country Club to finish at 27-under 189 through 54 holes.
According to the PGA, four amateurs have won a Tour event since 1940. The last amateur to win on Tour was Phil Mickelson at the 1991 Northern Telecom Open in Tucson, Ariz.
"I made a lot of putts," Dunlap said after the third round. "The putter felt really good. I don't think I missed anything that I should have made, per se... I don't have any negatives after that round. I did everything pretty well."
The 60 matches the amateur record set by Patrick Cantlay at the Travelers Championship in 2011. The 12-under score relative to par is a new amateur record.
After his round, Dunlap admitted that shooting a 59 did cross his mind.
"On 17, I did (think about it)," he said. "I ran it by, I wasn't leaving that one short. No - and (caddie) Hunter (Hamrick) mentioned it on 18. He's like, ‘Let's go ahead and make this.' So, no, I did."
Sam Burns, the leader after two rounds, shot a 7-under 65 on the Stadium Course and sits in second and 24-under 192.
Justin Thomas nearly matched Dunlap's mark, tying the Stadium Course record with an 11-under 61 to sit in third at 23-under 193. Thomas, who won a national title as a member of the Crimson Tide in 2013, equaled his career high with six straight birdies at one point.
Thomas and Dunlap will be in Sunday's final group, something Thomas knows will draw extra eyes in Tuscaloosa.
"I haven't played with Nick (on tour), no. We've just been in touch and we text from time to time." Thomas said. "But he's in college, so we don't run into each other too much. I know (Alabama golf coach Jay) Seawell will be a very, very happy man, and a very anxious man (Sunday), I'm sure."
South African Christiaan Bezuidenhout shot a 7-under 65 - also on the Stadium Course - to finish the day in fourth, two strokes behind Thomas.
A quintet of players finished Saturday tied for fifth at 20-under 196: Eric Cole (shot 66 on the Nicklaus Tournament Course), South Korean Si Woo Kim (66, Stadium Course), Adam Hadwin of Canada (66, La Quinta Country Club), J.T. Poston (64, Nicklaus Tournament Course) and Xander Schauffele (63, Stadium Course).
