The two Americans both threatened to match the course record set by Nick Price in 1986 and equalled by Greg Norman a decade later.
Simpson shot up the leaderboard after covering the first two days in one-under par to put himself in an excellent position going into the final round at nine-under.
The 2012 U.S. Open champion stumbled on the sixth hole, committing a "sloppy" bogey, but he recovered spectacularly by logging seven birdies over the next dozen holes.
"I just kind of had a talk with myself going back to seven tee thinking about all my bad shots this week have been just some poor thinking and not being really committed to what I decided to do," said Simpson.
"And I just said hey, if you stay fully committed the rest of the day, you're swinging great, you're putting well, you're going to make birdies -- and that's what I did."
Cantlay, who was languishing near the back of the pack at two-over after the first two days, roared back into contention to go into the final round at six-under par.
He recorded eight birdies and not a single bogey, by far his best Masters performance, having failed to make the cut in 2018 and finishing tied for 47th in 2017.
"I got out early and the ball was going the right distance, the wind wasn't blowing too much, greens were soft, and the hole locations were much easier than the last few days," said Cantlay.
"It was kind of the perfect storm for a good round."
(Reporting By Amy Tennery; Editing by Ian Chadband)
