The Swede started the day one shot clear of Mickelson and recovered from a bogey at the first to produce a stunning major-championship record-equalling round of 63, eight under par.
His effort featured 10 birdies and took him to a 20-under total to finally win his first major by a three-stroke margin with the lowest overall score in Open history.
Mickelson did very little wrong, with four birdies, an eagle and not a single dropped shot, but his 65 was not enough as he was left to settle for second place at 17-under.
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.
Emphasising the gulf between them and the rest of the field was the fact that JB Holmes of the United States took third place with a six-under-par total for the weekend.
“This will take a while to sink in,” said Stenson, at last a major winner at the age of 40, as he clutched the trophy.
He had previously endured seven top-four finishes in majors and was the runner-up to Mickelson in the British Open at Muirfield in 2013.
Coming into the day, the prospect of a two-way shoot-out for the Claret Jug evoked memories of the legendary 'Duel in the Sun' down the Ayrshire coast at Turnberry in 1977, when Tom Watson edged out Jack Nicklaus after the two Americans had separated themselves from their rivals.
And the showdown between the two forty-somethings more than lived up to its billing as they produced an exhibition of world-class golf.
Stenson recovered from dropping a shot at the par-four first by recording five birdies before the turn and also recording a birdie at the 10th.
There was a wobble when he missed a par putt at the difficult 11th hole but remarkably he produced three straight birdies from the 14th to put the outcome beyond doubt.
The game was up for Mickelson when Stenson came up with a stunning long putt for birdie at the 15th to go two shots clear and leave his opponent with too much to do.
The final gap became three shots as Stenson holed out for birdie at the last to follow Mickelson's feat in the first round on Thursday by shooting 63.
“It's probably the best I've played and not won,” admitted Mickelson.
“I'm disappointed to come in second but I'm happy for Henrik.”
Stenson's 20-under-par total of 264 was a record low score in The Open at Royal Troon, obliterating the American Justin Leonard's 12-under winning effort in 1997.
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 victory prevented Mickelson from becoming the seventh straight American winner of an Open to be held at Troon. The 46-year-old Californian was also denied the honour of becoming the oldest winner of the Claret Jug since Tom Morris Senior down the road at Prestwick way back in 1867.
Mickelson remains on five majors, despite having started his week with that remarkable 63 of his own.
On Thursday he was left cursing when a birdie putt at the 18th lipped out, denying him the first ever 62 in a major, and he was also agonisingly denied lifting the Claret Jug for the second time by Stenson's brilliance.
Both players put the so-called 'Big Four' of Jason Day, Dustin Johnson, Jordan Spieth and Rory McIlroy thoroughly in the shade on Scotland's west coast.
McIlroy, the 2014 Open champion at Hoylake, finished the highest of that quartet after an impressive 67 on Sunday left him tied for fifth at four under par overall alongside England's Tyrrell Hatton and Sergio Garcia of Spain.
Steve Stricker, another American, was fourth at five under par.