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Henley lit up the Golf Club of Houston course, firing 10 birdies in a seven-under-par 65.
His 20-under-par total of 268 tied the 72-hole scoring record established by Phil Mickelson in 2011 and gave Henley a three-shot triumph over overnight leader Kang Sung of South Korea, who had two birdies and two bogeys in his even-par 72 for 271.
"It hasn't quite hit me yet that I've earned a spot to the Masters," said Henley, who started the week ranked 117th in the world.
Americans Luke List and Rickie Fowler shared third on 272, List carding a final-round 68 as Fowler signed for a two-under 70.
Henley began the day four shots off Kang's lead. He birdied five of his first eight holes and by the time he was done his double-bogey from a bunker at the par-three ninth was barely a blip in his round.
"I tried to go into it with the mentality that I could do it and win it," Henley said afterwards. "And I started to believe it."
He had already pulled into a tie for the lead when he strung together three birdies in a row at the 13th, 14th and 15th. He tapped in to take the lead at 13 after lagging his first putt from 98 feet. He drained a 36-footer at the par-three 14th and tapped in again at 15.
Henley added one more birdie before a finishing bogey that made no difference as Kang was unable to find a birdie on the back nine.
"Russell just played it great," Kang said. "He putted really well, made almost everything. He's been putting really well since yesterday and last few days, and then I haven't been putting really well, not like the first two rounds. That was the big key, I think, the putting."
The third victory of Henley's career, and his first since 2014, earned him the final invitation to the Masters, the first major of the year that tees off on Thursday at Augusta National.
Fowler, whose third-round charge was derailed by a bogey, double-bogey finish on Saturday, fell out of contention with a double bogey and a bogey in his first four holes.
He had five birdies, including three on the back nine, but by that time Henley was out of reach.
Argentine veteran Angel Cabrera was never in contention, but he delivered the spectacle of the day with his hole in one on the 226-yard ninth hole, his last of the day.
Leading scores after Sunday's final round of the PGA Tour's Houston Open (par-72, USA unless noted):
268 - Russell Henley 67-67-69-65
271 - Kang Sung (KOR) 65-63-71-72
272 - Luke List 68-71-65-68, Rickie Fowler 64-71-67-70
275 - Daniel Berger 70-67-71-67
276 - Hudson Swafford 67-67-75-67
277 - Rafael Campos (PUR) 69-71-68-69
278 - Zac Blair 69-67-75-67, Kyle Stanley 66-71-71-70
279 - Jon Rahm (ESP) 71-72-69-67, Andrew Loupe 68-70-72-69
280 - Michael Thompson 70-73-69-68, Jason Dufner 68-69-73-70, Michael Kim 70-70-70-70
281 - Keegan Bradley 67-71-76-67, Justin Rose (ENG) 67-71-73-70, Andy Sullivan (ENG) 68-76-67-70, Jhonattan Vegas (VEN) 66-74-70-71, Aaron Baddeley (AUS) 68-68-73-72
282 - Sam Saunders 71-73-69-69, Robert Garrigus 68-73-71-70, Danny Lee (NZL) 69-69-72-72
283 - Spencer Levin 70-71-74-68, Geoff Ogilvy (AUS) 68-74-72-69, Peter Uihlein 71-71-72-69, Mackenzie Hughes (CAN) 72-70-73-68, D.A. Points 68-71-74-70, Charley Hoffman 71-72-69-71, Bernd Wiesberger (AUT) 70-71-72-70, Vaughn Taylor 67-74-75-67, Harold Varner 68-73-71-71, Andres Gonzales 75-67-75-66, Stewart Cink 66-71-73-73
