Fitzpatrick had six birdies and an eagle when he holed out from 149 yards on the par-4 third hole.
The Englishman finished the day at 14-under 199, poised for his first tour win since picking up his major title at Brookline, Massachusetts, 10 months ago.
"That's ideally the start you need on a Saturday, isn't it?" Fitzpatrick said of the unexpected hole out.
"It just gets you off to a good start."
Cantlay, who lost in a playoff to Jordan Spieth a year ago, shot 66 to move to 13 under and into position for another try at the winner's jacket.
Spieth is also in the hunt, another stroke back after a 66. He is seeking to do something he has not yet managed in his career - defend a title.
Also carding a third-round 66 was Australia's Cameron Davis, who sits tied for ninth at 10 under after 54 holes.
Adam Scott (68) is two shots further back in a tie for 21st, while Lucas Herbert (71) is three under overall.
World No.2 Scottie Scheffler (69) and second-round leader Jimmy Walker (72) are among five players tied for fourth at 11 under.
Walker, who opened with 65-65, reached 14 under and was tied with Fitzpatrick before blowing up with with three bogeys on the back nine.
Masters champion Jon Rahm gave spectators who have celebrated his second major victory all week something to scream about at Harbour Town when he opened with three straight birdies.
The world's No.1 player finished at 8-under after a 69 and is 18 holes away from a well-deserved rest after winning at Augusta National last week for his second major title.
Fitzpatrick has struggled much of the year, missing four cuts in seven events before arriving at Augusta National last week, where he tied for 10th.
"This is a golf course that's shown that there are low scores around here this week," said Fitzpatrick.
The 28-year-old, surpassed his previous tour low of 64, also accomplished at this venue two years ago.
Cantlay has had four top-seven finishes at Harbour Town in five previous appearances.
He had birdies on three of his first six holes to give himself a chance at another high finish.
"I've had a good track record around this golf course and had a number of chances, so should be nothing new tomorrow," he said.
