The South African lived up to his billing as the highest-ranked player in the field with straight birdies at the 18th hole, once in regulation and then twice in the play-off after the pair finished tied on 13 under par in regulation.
The last of those was enough for victory as Olesen, who had earlier overcome an early four-shot deficit on the front nine with a round of 68, could only manage a par after missing the fairway and being forced to lay up.
Coetzee, 28, started the day one shot ahead of compatriot and playing partner Thomas Aiken as well as Dane Olesen, but pulled clear with a sensational start as he birdied the second and third from close range.
Already a winner at his home Pretoria GC course this season in the Tshwane Open, the South African found himself three clear and maintained that advantage when he birdied the fifth.
Olesen, whose driving was all over the place on Saturday, had failed to get up-and-down at the first and was four behind at that stage but the 25-year-old???s hot putter helped him birdie the fifth from 12 feet, convert an eight footer at the sixth and make light work of the long seventh.
The same could not be said of Coetzee, who drove out of bounds on the par-five seventh and bogeyed, and Olesen, returning from a three-month injury lay-off, made a 10 foot putt at the ninth for a share of the lead on 13-under.
However, Olesen bogeyed the 10th to drop one behind, and with birdies on the back nine hard to come by the gap remained unchanged until he birdied the 17th from six feet, with Coetzee missing an almost identical-length par putt on the hole behind moments later.
Needing to birdie the last to force a play-off, world No 63 Coetzee almost went one better when his eagle effort from 30 feet finished inches short and he tapped in for a closing 69.
On the first trip back down the 18th Coetzee???s eagle attempt from 40 feet lipped out as both players birdied, but a third gain proved lucky as he secured a third European Tour victory and close on a place at next month’s US Open.
“I felt nervous during the day but I???m happy to have pulled through it and posted a good number,” said the victorious Coetzee. “Thorbjorn did unbelievable well to put pressure on me the whole day and I had to work hard to catch him. I???m just happy the job is done, it doesn???t matter how many holes it takes, you just get it done.
“I would have love to have won it on the first play-off hole with an eagle and do it in style, but I???m obviously happy just to have won.
“They???ve done a fantastic job with the tri-sanctioning of this event and bring us to a fantastic place like Mauritius. It is great to get another victory on the European Tour and obviously it counts towards my Sunshine Tour ranking as well. It is nice to finally get an Asian Tour win as well.”
Olesen added: “It???s disappointing, of course, when you are in a play-off and you don???t win. There were a couple of key moments out there where I hit a bad tee shot on the 10th and the 14th, and those two holes cost me. It???s difficult to play that 18th with a three wood, but I???ve been hitting my driver pretty bad so I had to.
“George hit some great shots there in the play-off to get two birdies, as it???s not an easy hole. So big congratulations to him and he???s played really well.
“I didn???t expect this before the tournament so to be in a play-off is unbelievable when I haven???t been in a tournament for three months. I???m very pleased with that and I can take it with me. It???s a good start.”
Singapore veteran Mardan Mamat, 47, birdied two of his last three holes for a 67 which left him third on 12 under par, with Aiken fourth on 11-under following a round of 70.