Defending champion Spieth had led the tournament by four after six holes of his final round but a disastrous run after the turn saw the American drop six shots in three holes, including an astonishing quadruple bogey at the par-three 12th when the 22-year-old twice sent his ball into the water.
That opened the door for Willett, the world number 12 to assume the lead. He had started the day three behind Spieth at level par but slowly chipped away until he took control at the 14th with his fourth birdie of the day.
Another birdie followed the par-three 16th and an excellent par at the 18th sent him into the clubhouse on five under par with a closing 67 to wait for Spieth's response over the final four holes.
He had been the last to arrive at Augusta National on Monday having delayed his travel following the birth of his first child, Zachariah. But in only his second Masters appearance he showed his comfort with the surroundings, with rounds of 70, 74 and 72 before his best of the week when it mattered most.
The Yorkshireman, 28, becomes just the second Englishman to win the Masters after three-time champion Nick Faldo, who last title here came in 1986, and the first European champion since Jose Maria Olazabal in 1999.
Having fallen from seven under at the ninth to one under after 12, Spieth did his best to rally with birdies at the par-five 13th and 15th to get to three under but he needed more and a missed eight-foot birdie putt at 16 spelled the end of his challenge. He bogeyed 17 and Willett could celebrate his maiden major victory.
Compatriot Lee Westwood had eagled the 15th to get to three under only to bogey 16. He signed for a 69 to claim joint runner-up spot on two under with Spieth, who signed for a 73, his third successive over par round of the week.
Another Englishman, Paul Casey, tied for fourth with Americans Dustin Johnson and JB Holmes on one under while another Yorkshireman, Matt Fitzpatrick also shot a 67 to share seventh place on level par with Irish Open champion Soren Kjeldsen of Denmark and Japan's Hideki Matsuyama.
World number three Rory McIlroy was part of a tie for 10th place on one over following a closing 71, his chance of securing the career grand slam gone for another year as he admitted the chase for a so-far elusive Masters title had become a mental issue.
He had been a shot behind 36-hole leader Spieth on Saturday morning only to card a third-round 77 and McIlroy said: “I was in a great position going into the weekend, a shot back in the final group on Saturday and I just didn't play the golf I needed to when it really mattered.
“That's the thing that I take away not just from this week, but from previous Masters. I've been in position before and I haven't got the job done when I needed to and I don't think that's anything to do with my game, I think that's more me mentally and I'm trying to deal with the pressure of it and the thrill of the achievement if it were to happen. I think that's the thing that's really holding me back.
“So, the more times I can get in position to win this tournament, the more times I'll learn and I'll know what not to do. And I feel like I learned a lot yesterday reflecting on it and that's something that hopefully I'll do things differently.”
McIlroy could take a lead from Willett, who only needed one invitation to make the most of his opportunity and seized it with both hands, then closed out the deal with nerveless composure over the closing three holes. He had a two-shot lead going to the last, then hit an excellent approach shot on the par-four from 166 yards out to 14 feet and two-putted from there for his 67 before retreating to the clubhouse to wait.
Earlier in the round, Ireland's other weekend competitor Shane Lowry along with Davis Love III and Louis Oosthuizen had all holed in one at the 16th.
Lowry's ace helped him to a closing 75, finished in a tie for 39th at 10 over par, his opening 68 followed by disappointment and frustration with rounds of 76 and 79 on Friday and Saturday.