The 30-year-old carded a 68 to finish at 16 under after 72 holes at TPC Southwind, with Matsuyama and Burns storming through the field on day four with rounds of 63 and 64 respectively.
Matsuyama almost won it on the first trip back up the last as he lipped out from 20 feet but Ancer put his approach to six feet at the second attempt before Burns incredibly got inside him.
Ancer piled on the pressure by making his putt and when Burns missed with his effort, Ancer had his first European Tour, US PGA Tour and World Golf Championships win.
American Harris English had entered the fourth round with a two shot lead at 18 under but 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.
Ancer first joined the European Tour in 2019 and his best finish coming into this week had also been at a World Golf Championships when he finished in a tie for fourth at that season's WGC-HSBC Champions.
He finished 62nd on last season's Race to Dubai Rankings in Partnership with Rolex but will now move up to seventh in 2021, as well as a career high 11th on the Official World Golf Ranking.
"I felt really good the whole week and I've been feeling good for a long, long time," he said. "I just needed to be patient and just keep putting myself in good positions to win the tournament.
"There's some tournaments I felt that I played good enough to win and had a great round and things just didn't go my way and I ended up not winning the tournament. So you've got to get lucky times to win events and I'm happy things went my way this week.
"I know Mexico is in a better spot than some years prior right now, Mexican golf, but I think it's getting a lot better. 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. I'm really pumped where things are headed in my country."
There have been some times when I thought I made enough birdies to win and I hadn't had a win so it's just kind of how it goes. I'm happy that I got lucky
Matsuyama started the day nine shots off the lead and went out alongside Burns a full hour before the leaders, with his charge looking unlikely after a slow start.
He made the most of the par five third but then recorded four pars before he almost aced the eighth and put an approach to 11 feet at the ninth.
Pinpoint approaches to the 12th and 13th were followed by a long putt on the 14th, and when he got up and down on the 16th, he was 16 under.
Burns' move up the leaderboard was more swift, as he holed a 15 footer on the sixth and put approaches inside ten feet at the second, fifth and ninth to turn in 31.
Another smart second brought a birdie on the tenth but he made a double bogey on the 13th after sending his ball out of bounds left, before he bounced back by sandwiching an up-and-down on the 16th with a pair of 25 footers on the 15th and last.
English lost his overnight lead on the first as he came up short and failed to get up and down but a two putt birdie at the par five third was followed by an approach to four feet at the sixth, with a 30 footer at the eighth briefly handing the 32-year-old a three shot lead.
That advantage was trimmed to two at the turn and while English made a double bogey after finding water off the tee at the 11th, he still led by two as closest challenger Bryson DeChambeau made a triple on the same hole having bogeyed the tenth.
But when the leader found more water off the tee at the 14th, there was a five way tie, with Ancer also at 16 under.
The man with dual Mexican and American citizenship had hit a stunning approach to the second and birdied the third from the fringe but he dropped a shot after coming up short at the sixth before picking it back up on the 13th with a second to four feet.
English then got in trouble down the left on the 16th but sent his third right and when he failed to get up and down, he was out of the lead.
Smith had been in line for a play-off place but a double bogey on the last saw him sign for a 72 and slip back alongside Berger, who finished with a 66, and Casey, who carded a 67.