Nineteen of the Russian's 21 championships have been won on hard surfaces
A resurgent Medvedev secured his first trophy for over two years with success over Frenchman Moutet in the tournament final.
The Russian, a former world number one, has endured a difficult year, winning just one match at the big events and experiencing a setback at the New York tournament.
The player's varied form on the ATP Tour - and issues with his service game in especial - caused him falling of the world's top 10 and eventually parting ways with former coach Gilles Cervara.
Currently working with his coach and Goetzke, the tennis star has consistently improved in the recent weeks - culminating in a closely contested competitive win over opponent Moutet.
It is Medvedev's first trophy after the 2023 Rome Masters and initial on a hard court from spring of that year.
The win also signifies Medvedev has secured twenty-one titles at 21 distinct tournaments.
"This marks the first occasion I am participating in a event with my both daughters together, so to secure a trophy is especially meaningful," commented world number 14 the champion.
"This championship is in honour of my second daughter her name, because the first title I won when my first daughter Alisa was delivered was for her."
Medvedev, the 2021 US Open champion, had lost his past six finals, including in Halle previously this season.
His retrieving approach and regular emotional displays on the playing surface had caused commentators to question if his capacity to perform at the top level was over.
But the 29-year-old has made changes in recent weeks, striving to come to the net and play assertively, backed up by an better serve.
The champion recovered from a deficit in the opening set against the tricky Moutet, taking four of the concluding five exchanges to gain the advantage.
The competitors swapped service breaks in the following set before Moutet, striving for his maiden tour championship, took control to force a decider.
He defended four early break points in a close final set before the champion broke to that score to serve for the match.
Suitably, he earned victory with a rally at the net area, with his opponent unable to handle a volley after two hours and 38 minutes.
