He finished at 20-under 260, four clear of Englishman Eddie Pepperell who carded a final-round 64.
Stone and Pepperell secured late exemptions into the British Open starting at Carnoustie on Thursday.
Overnight leader Jens Dantorp of Sweden clinched the third Open berth up for grabs when he tied for third on 15 under with South African Trevor Immelman and American Luke List.
"It's been a long 18-month journey, making a few changes, but the swing felt incredible today, the putting felt even better and the mental state was flawless," said the 25-year-old Stone who arrived in Scotland with no form to speak of this season.
"A day shooting 60 and winning ... is something I'm going to hold dear to my heart for a very long time," he added.
"Hopefully, I can find some accommodation (in Carnoustie). I wasn't exactly planning on going through."
Stone's final margin on another day of low scoring on the bone-dry links course disguised the close battle for most of the final round, with the winner only breaking clear when he sank a 30-foot putt to eagle the par-five 16th.
History beckoned when he stuck his approach to about eight feet at the last, but his putt for 59 broke left to narrowly miss the hole, leaving Stone in the unusual position of looking dejected despite his dominant performance.
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