Monfils, ranked 140th, won 4-6, 7-6(6), 6-3 against his 109th-ranked opponent for his second title in the Swedish capital after 2011.
The 37-year-old has endured a season plagued by foot and wrist injuries.
He is just the fourth player above the age of 37 to win an ATP Tour title since 1990, joining Roger Federer, Ivo Karlovic and Feliciano Lopez.
Monfils was pushed hard in the two hour-35 minute clash on his way to becoming the oldest champion in the history of the tournament.
"Credit to Pavel," said Monfils after the first meeting between the pair.
"He has had an unbelievable week and he was pushing me on the court. I ran a lot today. When I had to save break points in the second set, I tried to push through.
"I don't know how I pulled up but I kept believing and fighting."
Monfils dropped just two sets on his way to the trophy, his first since Adelaide in January last year, and will move up to 89th in the ATP rankings.
Kotov, 24, was aiming to win his first title on Sunday in his maiden final.
