Nineteen of Medvedev's twenty-one championships have been won on hard courts
A resurgent the Russian player won his maiden title for 882 days with success over opponent Moutet in the Almaty Open final.
Medvedev, a ex- number one, has endured a challenging year, winning just one match at the major tournaments and experiencing a difficult moment at the American major.
The player's varied form on the ATP Tour - and struggles with his serving in specific - caused him dropping out of the world's top 10 and eventually separating with long-term mentor his previous coach.
Currently collaborating with Thomas Johansson and Rohan Goetzke, he has gradually advanced in the past month - climaxing in a closely contested three-set victory over opponent Moutet.
This represents the player's first trophy following the Rome competition and initial on a hard surface since March of the previous year.
The win also means he has won 21 titles at 21 different tournaments.
"It's the first time I have attended a event with my both daughters at once, so to secure a championship is particularly special," commented tennis professional the champion.
"This victory is in tribute of my child Victoria, because the first title I got when my first daughter her name was born was for her."
Medvedev, the past champion, had been defeated in his last six championship matches, including in Halle earlier this year.
Medvedev's defensive strategy and frequent emotional displays on the playing surface had led commentators to wonder if his capacity to compete at the elite level was finished.
But the athlete has made changes in recent weeks, attempting to approach the net and compete aggressively, supported by an better serve.
Medvedev came back from a break down in the first set against the difficult opponent, securing four of the last five exchanges to go ahead.
The competitors exchanged service breaks in the second set before the Frenchman, striving for his first tour title, seized the initiative to push for a third set.
Moutet rescued four initial break points in a tight third set before the Russian won the break to that score to close out the victory.
Suitably, the winner clinched the title with a shot at the net area, with Moutet unable to manage a shot after two hours and 38 minutes.
