The Frenchman was leading 6-3 2-6 6-4 4-2 when his opponent withdrew.
Despite some good serving early on, Tursunov lost the opening set 6-3 due to the number of unforced errors coming from his racket.
He recovered in the second set, dominating the majority of the rallies with his forehand from the back of the court.
Both men continued to attack in an enthralling third set, but it was Gasquet who just managed to keep the ball in the court more often and edged ahead.
That was enough to break the Russian's will and Tursunov went a break down early in the fourth, before retiring at the start of a Gasquet service game.
The number eight seed will now face Milos Raonic in round four.