The Frenchman lost the first set and fell 4-3 behind in the second before rallying to win 4-6 6-4 6-4 and claim his third tournament win of the year over the Kazakh qualifier.
The victory also moves Gasquet up to ninth place in the Race To London - which would be enough to qualify for the season-ending Barclays ATP World Tour Finals in light of Andy Murray's withdrawal.
Gasquet looked down and out when world number 102 Kukushkin grabbed the decisive break to win the first set and took command in the second.
But the 27-year-old Gasquet fought back to level the score before breaking twice in the decider to claim victory in two hours and 24 minutes.
Earlier, Romania's Simona Halep won the women's title with a 7-6 (7/1) 6-2 win over Sam Stosur.
Halep has now won five WTA titles this year - second only to Serena Williams - after prevailing in a match lasting one hour and 41 minutes.