Medvedev, the 2021 US Open champion, had lost his past six finals, including in Halle earlier this year.
His defensive play style and frequent outbursts on court had led some to wonder if his ability to compete at the top level was over.
But the 29-year-old has made changes in recent weeks, making an effort to come to the net and play aggressively, backed up by an improved serve.
He came back from a break down in the first set against the tricky Moutet, winning four of the final five games to take the lead.
The pair exchanged breaks in the second set before Moutet, bidding for his first ATP Tour title, took control to force a decider.
Moutet saved four early break points in a tight third set before Medvedev broke to 15 to serve for the match.
Fittingly, Medvedev secured victory with a point at the net, with Moutet unable to control a volley after two hours and 38 minutes.