Stenson becomes the first Scandinavian man ever to win a major and the first player from continental Europe to lift the Claret Jug since the late, great Seve Ballesteros of Spain in 1988. His final-round 63 gave him a total of 264, 20 under for the tournament.
Both enjoyed cast-iron pars at nine to turn in 32, and their remarkable form continued at the 10th - one of the toughest holes on the course - as Stenson holed from 12 feet for birdie before Mickelson followed him in. His 20-under par matched Jason Day's record for lowest under par at last year's PGA Championship.
"I didn't know until they told me in the recorder's hut", Stenson said when asked if he knew that last putt was for 63. Woods did it in 2000 at St. Andrews.
Mickelson remains on five major titles, including one in The Open at Muirfield in 2013 when he beat Stenson to the trophy.
Maybe he can take that silver jug down to Rio in his search of Olympic gold.
It was the first time all day the two had been separated by more than a single shot - they were tied on eight holes, Stenson led by one on five others - and there was no way back.
Mickelson narrowly missed an eagle putt on the 16th hole, and Stenson kept his two-shot lead with an up-and-down for birdie from deep rough.
He gave Sweden a long-awaited major in men's golf, 19 years after Jesper Parnevik lost a 54-hole lead at Royal Troon.
"I had the best seat in the house".
"I think that's probably why it's disappointing in that I don't have a point where I can look back and say, "I should have done that or had I only done this".
The Swede won his first major with precision.
"I'd like to thank Phil for a fantastic battle, we played some great golf and I am delighted to come out on top but thank you very much for a great fight".
"I'm really proud to have done that, and it's going to be massive for golf in Sweden with this win", Stenson said. He also would be the third-oldest major champion behind Julius Boros (48) and Morris, with whom Mickelson shares a birthday - June 16, albeit 149 years apart. "But I'm happy with everything".
Other than Mickelson, the nearest player to Stenson was J.B. Holmes, 14 strokes back. He will surely be a force to be reckoned with when the USPGA gets underway at Baltusrol in 10 days' time. But it seems there have been a couple of putts like that this week.
On a surprisingly bright and calm afternoon on Scotland's west coast, it was set up for a shoot-out between the two players and they did not disappoint. That run ended Sunday, when he shot a 3-under 68 in the final round at Royal Troon.
Spieth hopes to build on that momentum at the PGA Championship in two weeks. In the elevated quality of the twosome, Stenson became the first Swedish male major champion through near-spotless golf, with a whopping 10 birdies to offset two three-putt bogeys from outer edges of greens.