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The American world No 1 secured his seventh win of the year with a course record-equalling 62 at Le Golf National to edge out silver medallist Fleetwood on 19-under-par for the tournament.
Japan’s Hideki Matsuyama had to settle for bronze on 17-under after a dramatic final round that saw several players challenge for the medals.
Jon Rahm sat four shots ahead after 10 holes but fell away badly to finish tied-fifth with Rory McIlroy, whose hopes were dashed by a double-bogey on the 15th.
Reigning champion Xander Schauffele, who was tied for the overnight lead with Rahm, dropped out of contention with a three-over back nine.
Frenchman Victor Perez delighted the home fans with a stunning surge through the field but finished in fourth place despite his 63, which included a run of four birdies and an eagle in the space of five holes.
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