Fleetwood shot a five-under-par 67 to finish on 11 under, one ahead of Fox, who would have leapfrogged Rory Mcllroy at the top of the DP World Tour Rankings in Partnership with Rolex with a victory.
By that time Højgaard was one clear having birdied the par-five second to hit double digits in the red.
Fleetwood got his round going with a gain at the second and two in a row from the fifth before the storm warning led to a delay of more than three hours.
Højgaard, shortly after the restart, walked off the seventh with a bogey after missing the green left with his tee shot, and another dropped shot at the eighth before a double at the ninth, having missed the fairway with his drive and putting his third in the water, left him too much to do.
Fleetwood and Sharma took advantage, the pair moving to ten under with birdies at the par-five ninth. Sharma posted another at the tenth while Fleetwood could only par as Fox joined the party.
Fox had started with six straight pars then birdied four in a row. He left himself around eight feet at the seventh and eighth, was unlucky with a long eagle putt at the ninth before tapping in, and then holed an 11-footer at the tenth.
Fleetwood slipped back after a bogey at the 12th but a moment of magic at the 14th turned things back in his favour.
He was in the bunker left of the par-five in two and the long shot from the sand was far from a guaranteed up and down. But he holed it for eagle it to jump to 11 under and moments later, his playing partners Sharma and Fox rolled in their birdie putts and the three of them walked to the 15th tee sharing the lead.
Sharma's challenge ended with bogeys at the 16th and 17th and when Fleetwood tapped in for par at the last, after almost sinking a huge birdie attempt, Fox paid for a poor tee shot and posted his only bogey of the day at the worst possible moment.
The victory was Fleetwood's sixth on the DP World Tour and first since this event three years ago, the last time it was staged due to the pandemic.
He said: "It's been a great week. I already had so many amazing memories of this place, such a great connection to it.
"It's amazing we got four rounds in in the end and it was such a lovely feeling coming down the back nine with a chance. You draw on the good memories that you have round somewhere you've won. I felt good all day."
On his bunker shot at the 14th, he said: "It wasn't the best lie but I felt okay. It wasn't a terrible place to be. I don't know how far it would have gone past but it came out lovely and went in. That spark, those kind of things happen."
Fleetwood had been ill earlier in the week and his participation had been in doubt.
He added: "It was touch and go on Thursday morning (whether he would play), then Saturday when we were up early (to finish the second round) I felt like I had nothing in me. Really poorly again.
"The doctor here has been amazing. I just gave his little son the winning golf ball. Without him there was not a chance on Thursday I'd have played.
"We didn't know what was happening whether it would be three rounds, two, so I just thought if I could keep moving, keep playing... and I was playing well.
"It wasn't overly stressful for me and I just had in my mind to keep going... and you never know what happens."