Horschel came into the tournament having missed four consecutive cuts but turned his fortunes around with some incredible putting this week at the TPC Four Seasons resort near Dallas.
Australia's Day missed a short putt on the 18th hole that would have extended the playoff to hand Horschel the victory.
"That's not the way I wanted to win it," said Horschel. "Four missed cuts to come in here and I played this well -- someone is looking out for me."
American Jason Kokrak finished fourth on 270.
Horschel had the top putting percentage this week while world number four Day had made 95 percent of his putts inside five feet this season before missing what appeared to be a routine four-footer to drop the playoff.
"When you're in the position like that, you have to go for it and try to win," Day said.
Day, Horschel and Hahn all played in the final group, creating a match-play type scene.
Horschel's two-putt birdie on the par-five 16th moved him back into a tie for the lead with Day while Hahn's par on the hole all but ended his chances. Both Day and Horschel parred 17 and 18.
"I came here without any type of momentum, and the only thing I can hang my hat on was my practice sessions for the last several months have gone well leading to the tournaments," Horschel said.
Canada's Nick Taylor (65), Joel Dahmen (67), Matt Kuchar (69) and Cameron Tringale (72) were another stroke back.
World number one Dustin Johnson never made a serious run and finished tied for 13th at 274.
Defending champion Sergio Garcia, who won his first major title last month at the Masters, shot 74 and finished in a tie for 20th.
Disclaimer: No Business Standard Journalist was involved in creation of this content