It was the fourth ATP title and third on clay for 21-year-old Fils, who returned from a nearly eight-month injury layoff in February.
In the first game, he missed the baseline with a couple of shots, handing Rublev a break. But Fils dug deep to get even, breaking back in the fourth game to make it 2-2.
Rublev struggled to keep up as Fils, now more precise with his forehand returns, picked out the edges of the sideline and sent powerful returns past the frustrated fifth seed to win the first set in just 31 minutes.
Fils won seven games straight before Rublev managed to hold his own serve to make it 1-1 in the second set. The 28-year-old fended off seven break points to prevail in a 13-minute battle to stay even at 2-2.
Fils outwitted Rublev with a backhand drop shot to get a decisive break and go 4-2 up. But when he was 5-3 up serving for the tournament, the world number 30 made three straight errors to hand Rublev a break.
Rublev then saved three match points to hold his serve and make it 5-5, and broke Fils to go 6-5 up. But Fils broke back in the very next game to force a tiebreak, which he won 7-2.