The third-seeded Italian clinched his seventh title overall, all of them on outdoor clay.
No. 4 Gasquet was also going for his second ATP title of 2018.
Fognini won the Brazil Open in March, and Gasquet won on grass at s-Hertogenbosch last month, the Frenchman's 15th title.
Fognini won five straight games in the opening set after conceding the first break of the match. Three straight double-faults let Fognini down in the second set as Gasquet went on to force the decider.
Fognini won with his fourth championship point as Gasquet's forehand went long.