The 26-year-old American notched his second victory in three starts this season and the fourth of his young career, having shot a seven-under-par 64 then waited for his playoff chance at TPC Scottsdale's Stadium Course on Sunday.
Gotterup and Matsuyama ended up at 16-under 268 on a day when world No.1 Scottie Scheffler also made a big move and created considerable buzz.
Gotterup, who also won last month's season-opening Sony Open in Hawaii, posted nine birdies including five of the final six holes in the last round.
"I feel confident in what I'm doing and feel like I have played well enough to feel confident to be able to be in those (winning) positions," Gotterup said. "So far, I've been able to capitalise on those."
Matsuyama's erratic driving finally hurt him late in his final round. His tee shot on the 18th hole ended up in a bunker and he made a costly lone bogey in his 68.
Then Matsuyama's tee shot on the playoff hole landed in the water alongside the fairway. Gotterup played it steady and recorded another birdie to seal the deal.
Two of Matsuyama's 11 victories on the PGA Tour have come in this tournament. He also won in 2016 and 2017, but he couldn't finish the quest this time.
Overnight leader Matsuyama was trying to win a fourth PGA Tour event in a 24-month period. This marked the sixth time that Matsuyama has been the outright leader through 54 holes on the PGA Tour, and he had won each of the previous five times.
"I wanted to avoid the playoff as much as I could, but I just hit a bad tee shot there in regulation at 18," Matsuyama said.
Two-time Phoenix Open winner Scheffler, who began his season two weeks earlier by winning The American Express, became the biggest story of the day for much of the round. He was within one shot of the lead through 15 holes, though he was playing several groups ahead of the last threesome. He finished with 64.
Michael Thorbjornsen (67), Akshay Bhatia (67), South Korea's Si Woo Kim (68) and Denmark's Nicolai Hojgaard (68) shared third place with Scheffler at 15 under.
Scheffler played the front side in 3 under and then strung together three more birdies on Nos. 13-15. That included sinking a shot from the fringe about 72 feet away on the par-4 14th. He also birdied No. 17.
With Scheffler's round complete, Thorbjornsen briefly catapulted into the lead with an 11-foot eagle putt on No. 15. He gave one of those strokes back with a bogey on the next hole after launching his tee shot beyond the green on the par-3 hole.
Australia's Min Woo Lee dropped away as he posted five bogeys in a closing 73 to be tied 28th at seven under while countryman Cam Davis was last among players to make the cut after a closing 80.
