Olesen, who recorded his third European Tour victory, led by three shots going into the final round but a double bogey on the second hole and a bogey on the third saw him pegged back.
He responded immediately with birdies at the fourth and fifth, and never looked back as he avenged his runner-up spot at the pro-am event three years ago.
The Dane fired a one-under-par 71 to finish at 18-under 270 for the week, two clear of Americans Brooks Koepka (67) and Chris Stroud (68).
"I got too aggressive on the second hole. I hit a great drive down there and felt like I had a good yardage. I was just too aggressive, and as soon as you get too aggressive on a links golf course, you get punished," Olesen said.
"I just kept myself together and knew that I have tried it before, so stayed very patient.
"Finishing off at St.Andrews is a great feeling. I kept believing I would be back here again. It was nice this week just to play three great rounds of golf, then hold it together in the last round."
Olesen's win lifts him to 25th place in The Race to Dubai standings and he will also enter the top 75 in the world rankings.
(Writing by Mark Meadows; Editing by Mark Lamport-Stokes)
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