After receiving a bye into the second round, world No 1 Djokovic found himself up against plucky lucky loser Moutet, who beat Dusan Lajovic in his opening match.
Moutet brought that form with him into his first meeting with Djokovic, breaking in the seventh game to the delight of the home crowd.
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The 20-year-old, who reached the top 100 in June before beating Grigor Dimitrov at Wimbledon, then saved four break points to keep Djokovic at bay.
But after squandering two set points on Djokovic’s serve, Moutet’s resolve cracked when the Serbian broke to 15.
The tie-break then swung Djokovic’s way early on, and the 32-year-old closed it out 7-2.
Djokovic, playing in his first match since losing to Stefanos Tsitsipas in Shanghai three weeks ago, raced into a 4-1 lead in the second set.
And though he dropped serve once, Djokovic was not pegged back by Moutet as the Frenchman valiantly bowed out after almost two hours on court.
Djokovic faces Kyle Edmund in the next round after the Briton beat 14th seed Diego Schwartzman 7-5 6-3.
Rafael Nadal enjoyed a somewhat more straightforward win, with the second seed overcoming Adrian Mannarino 7-5 6-4.
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