Fleetwood held firm during the playoff on the par-five 18th, posting a birdie after Gallacher had missed a birdie putt from seven feet and Gonzalez three-putted from the back of the green.
It was Fleetwood's third appearance at the event and the win moves him up to 165 in the official world golf rankings.
(Reporting By Mark Pangallo; Editing by Sonia Oxley)
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