Nineteen of the Russian's 21 victories have been won on hard surfaces
A comeback Medvedev secured his maiden trophy for nearly 30 months with success over opponent Moutet in the Almaty Open final.
The Russian, a ex- world number one, has endured a torrid period, achieving just a single victory at the major tournaments and undergoing a difficult moment at the New York tournament.
The player's unpredictability on the ATP Tour - and struggles with his serving in especial - led to him dropping out of the elite rankings and ultimately separating with long-term trainer his previous coach.
Now working with his coach and Goetzke, he has steadily progressed in the last few weeks - peaking in a tightly fought competitive triumph over opponent Corentin Moutet.
This marks the player's maiden title since the Rome competition and maiden on a hard surface from spring of 2023.
Success also signifies Medvedev has achieved 21 championships at twenty-one different events.
"It's the first time I am participating in a tournament with my both daughters at once, so to win a championship is really nice," stated world number 14 the champion.
"This championship is in dedication of my second daughter my daughter, because the maiden victory I got when my older daughter Alisa was born was for her."
The athlete, the 2021 US Open champion, had failed in his last six finals, featuring in the German tournament previously this season.
Medvedev's retrieving strategy and regular emotional displays on the playing surface had prompted commentators to doubt if his ability to perform at the elite level was over.
But the athlete has made changes in the last few weeks, making an effort to approach the net and perform aggressively, reinforced by an improved serve.
Medvedev fought back from a deficit in the first set against the tricky Frenchman, taking four of the last five points to take the lead.
The competitors swapped breaks in the following set before the Frenchman, bidding for his first ATP Tour title, took control to push for a final set.
The opponent defended four early-game opportunities in a tight third set before Medvedev secured a break to 15 to serve for the match.
Suitably, the winner earned the title with a shot at the front of the court, with Moutet failing to control a return after two hours and 38 minutes.
