Gotterup, who followed up a first-round 63 with rounds of 71 and 70, said: “I didn’t think this would happen, especially after Friday and Saturday, but here we are.
“Hideki was playing great. I just figured I would stay loose.”
Victory gave Gotterup his the fourth PGA Tour title, and his second of the season after the Sony Open in Hawaii last month.
World number one Scottie Scheffler missed out on the play-off by one shot after going round in 64.
“I played pretty well – only one round where I didn’t have my best stuff,” said American Scheffler, who carded a first-round 73.
“If I get in the house the first day with a couple under par, it’s a little different story today.”
England’s Matthew Fitzpatrick finished in ninth on 13 under after a 69.