Cantlay birdied the 17th after his drive bounced over a fairway bunker and although he did find sand on the 18th, the world number four safely reached the green and lipped out from long range for another birdie.
The tap-in par gave Cantlay a final round of 69 and a winning total of 14 under par, with fellow American Scott Stallings also carding a 69 to finish 13 under.
"I hit a lot of solid shots and then I got a couple of good breaks," Cantlay said in a post-round interview broadcast on Sky Sports.
"Obviously on 17 that break is something I'm not expecting and it was really big for me to take advantage of it. I played a lot of great golf this week and I'm happy to come out of here with the win."
Cantlay's triumph means he will start next week's season-ending Tour Championship in second place in the FedEx Cup standings and on eight under par, two shots behind leader Scottie Scheffler under the controversial handicap system introduced in 2019.
Scheffler finished in a tie for third with Olympic champion Xander Schauffele on 11 under.
Rory McIlroy will tee off in Atlanta on four under after a closing 69, which included birdies on his last three holes, gave the four-time major winner a share of eighth place at Wilmington Country Club.
However, McIlroy's Ryder Cup team-mate Shane Lowry agonisingly missed out on a spot in the 30-man field at East Lake by a single place after a final round of 68.
The former Open champion looked set to qualify after his fifth birdie of the day on the 14th, but three-putted the 17th for a costly bogey.