The American, who won last week in Greensboro just to qualify for the playoffs, became the first over-40 winner of consecutive PGA Tour events since Fiji's Vijay Singh in 2008.
"You work hard no matter what, whether you are fighting something or you're playing great, because you never know when it can turn," Glover said.
"It has turned very quickly for me and luckily I've been in a good frame of mind to take advantage of it."
Glover, the 2009 US Open champion, sank several dramatic long putts on the back nine while Cantlay, the 2021 playoff champion, charged late to force a playoff, which began at the 18th hole at TPC Southwind in Memphis, Tennessee.
Cantlay's tee shot went into water left of the fairway. He blasted his third onto the green just inside 22 feet and just inside where Glover's ball sat after two shots.
Glover two-putted for par, leaving Cantlay a must-make putt. Cantlay's ball rolled past the right edge and Glover had the victory.
"I was fortunate to get into a playoff," Glover said. "I said the guns would be coming and they came and I was just last man standing this week."
Cantlay fired a six-under-par 64 and Glover shot 69 to each finish 72 holes on 15-under 265.
Glover won his sixth career PGA title and became the first over-40 playoff winner since Tiger Woods in 2018.
The 70-player showdown advanced the top 50 in season points to next week's BMW Championship, from which only the top 30 qualify for the Tour Championship in two weeks at Atlanta.
Four-time major winner Rory McIlroy, last year's playoff champion from Northern Ireland, fired 65 to share third with England's Tommy Fleetwood on 266. American Taylor Moore was fifth on 268.
Second-ranked McIlroy would have overtaken world number one Scottie Scheffler with a victory.
Cantlay chipped in at 10 to pull within two of Glover, who sank a clutch seven-foot pat putt at 11.
Cantlay made an eight-foot birdie putt at 15 to move within one but Glover answered with a tense 20-foot par putt at the 13th.
Glover then sent his tee shot at the 14th into water, falling into a share of the lead despite salvaging bogey on a 30-foot putt.
Cantlay seized the lead with a birdie at the par-5 16th while Glover missed a seven-foot birdie putt at 15.
Glover's tap-in birdie at 16 gave him a share of the lead but he made yet another intense par putt from just inside 12 feet at 17 and parred 18 to force the playoff.
A storm struck just before 54-hole leader Glover and Moore were due to tee off, bringing a 97-minute delay before play resumed.
Moore, one adrift to start, found a fairway bunker at the first, a bush way left off the second tee and water at the fourth on the way to bogeys to fall back.
Glover birdied the par-5 third, boosting his lead to two, the margin he enjoyed at the turn.
Matsuyama advances
Glover began the day 49th on the points list but easily advanced.
Japan's Hideki Matsuyama, the 2021 Masters champion, had three birdies and an eagle in the last six holes to shoot 65 and advance, jumping from 57th to 47th in points.
"All I knew is I had to make birdies, so I just put my head down and did the best I could," Matsuyama said.
Australian Cam Davis jumped from 62nd to 45th to qualify for the BMW by ending on 269 after a closing 67.
American Nick Hardy and Canada's Mackenzie Hughes fell out of the top 50 to miss out.
