Henley, 27, who hails from Macon in Georgia began the day four shots behind overnight leader Sung Kang of Korea.
But he shot a seven-under par final round, including 10 birdies for a third PGA Tour win.
England's Justin Rose and Andy Sullivan both finished on seven under in a tie for 15th.
The Masters begins on Thursday with Henley appearing at the event for the fourth time.
"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.
"I tried to go into it with the mentality that I could do it and win it. And I started to believe it."
Three consecutive birdies from the 13th gave him the key advantage and even a bogey at the final hole could not spoil the day as Kang failed 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 the 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."
America's Russell Henley clinched the final place at this week's Masters in Augusta after a three-shot victory at the Houston Open.