Fleetwood produced a sensational back nine, making six birdies to come home in 30 shots. The European No. 1 had started the day 2 behind overnight leaders Fisher and Thomas Pieters.
Fisher seemed to be on his way to a first European Tour victory for nearly four years after picking up four shots in the outward nine.
But Fleetwood birdied the 10th, 12th, 13th, 15th and 16th holes and closed out victory on the last when he rolled in a 5-footer.
Fisher needed an eagle on the last to tie, but he could only manage a par for Fleetwood to celebrate his fourth European Tour win.