19 of Medvedev's 21 victories have been won on hard surfaces
A resurgent the Russian player secured his first title for nearly 30 months with a win over opponent Moutet in the Almaty Open title decider.
The Russian, a former world number one, has endured a torrid period, winning just a single victory at the major tournaments and suffering a meltdown at the American major.
The player's inconsistency on the ATP Tour - and difficulties with his service game in specific - led to him dropping out of the top ten and ultimately parting ways with long-time coach Gilles Cervara.
Now working with his coach and Rohan Goetzke, he has steadily advanced in the last few weeks - culminating in a hard-fought three-set win over opponent Moutet.
It is Medvedev's initial title after the 2023 Rome Masters and first on a hard court after that period of that year.
The win also means he has secured 21 championships at 21 various events.
"For the first time I'm at a tournament with my both daughters simultaneously, so to get a championship is particularly special," said world number 14 Medvedev.
"This victory is in tribute of my second daughter Victoria, because the initial championship I got when my first daughter my daughter was delivered was for her."
The tennis star, the former major winner, had lost his past six finals, including in Halle in the past months.
His counter-punching play style and regular outbursts on court had led some to doubt if his skill to play at the highest level was over.
Yet the player has made changes in recent weeks, striving to approach the net and compete positively, backed up by an improved serving game.
The champion came back from a break down in the first set against the tricky Frenchman, winning four of the concluding five exchanges to take the lead.
Both players swapped breaks in the middle set before the Frenchman, bidding for his initial tour title, seized the initiative to force a decider.
Moutet defended 4 early-game break points in a tight final set before Medvedev broke to that score to serve for the match.
Fittingly, he earned the win with a rally at the front of the court, with his opponent failing to handle a return after two hours and 38 minutes.
