Ancrer, 30, carded a closing 68 to finish on 16-under-par after 72 holes at TPC Southwind in Memphis, with Matsuyama and Burns matching that total after storming through the field with rounds of 63 and 64 respectively.
Masters champion Matsuyama almost won it on the first trip back up the last as he lipped out from 20 feet before Ancer holed from six feet for a birdie at the second extra hole then Burns lipped out from inside him.
“I was trying to kind of let it all sink in there in the ceremony on 18,” said Ancer of a success that lifted him 12 spots to 11th in the world rankings.
“It was really surreal, something that I've been working for since I was a little kid. Definitely a dream, a dream come true to win on the PGA Tour, and to do it on a big stage like a WGC event was really, really cool, man.”
American Harris English had started the day with a two-shot lead, but he fired a 73 to finish a shot out of the play-off, one clear of countryman Daniel Berger, England's Paul Casey and Australian Cameron Smith.
Bryson DeChambeau's 74 left him at 12-under alongside fellow American Will Zalatoris, a shot clear of England's Ian Poulter and former world No 1 Dustin Johnson.
“It was a crazy round,” added Ancer. “I thought I was going to need a really low one to have a chance today, but it just worked out that it was like pretty much survival mode on the back nine.”
Lorena Ochoa won two majors and also enjoyed a spell as world No 1 in the women’s game, but Ancer is now leading the way in the Mexican men making an impact at the highest level.
“Hopefully this week will inspire some kids to get out there and play golf and learn about the game and just grow the game in general,” he said.
“I'm really pumped where things are headed in my country. Carlos (Ortiz) has been playing some really good golf as well. I feel like we've done a really good job. There's a lot more to come.”
MacIntyre finished in a tie for 15th - his best performance in a WGC stroke-play event - after signing off with a second successive 68 for eight-under.
The effort came on the back of the young Scot finishing in the top 10 in last month’s Open at Royal St George’s.
Due to Poulter jumping above him, the effort wasn’t enough to stop MacIntyre from dropping a spot to in the world rankings.
Martin Laird, the other Scot in the field in Memphis, is down to 118th in the standings, slipping two spots, after a closing 77 left him having to settle for a tie for 57th.
Meanwhile, South African Erik van Rooyen is up to 79th from 113th after he landed a maiden PGA Tour triumph with victory in the Barracuda Championship in California.
The 31-year-old finished with 50 points in the modified Stableford scoring system, winning by five points from American Andrew Putnam.