The 28-year-old, needing a par on the final hole, held his nerve despite pulling his drive way to the left.
He recovered with an excellent up and down from a greenside bunker to complete a final round of 64 and 22-under-par total to claim the fifth European Tour title of his career.
Italian Molinari, who last week won the PGA Championship at Wentworth, put the pressure on with a birdie on the final hole to the delight of the home crowd.
He finished one shot behind Olesen on 21 under par with Slattery a further shot adrift in third. The Englishman paid for bogeys on the second and 14th holes and his eagle on the 15th was not enough to put him back in the lead.
Another Englishman, Danny Willett, returned to form with a final-round 69 to finish joint eighth, one of six players on 16 under par.
Olesen said: "It's been a little bit of a tough time for me this year, so to come here and win was amazing, especially coming from behind as I haven't done that before.
"I had a couple of good up and downs, there was a couple of tee shots I don't really like on this course, 15 and 16 I've struggle all week with them. But it's been good."
(Writing by Brian Homewood)
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