Gasquet broke his opponent in just the third game of the opening set thanks to a couple of forehand winners.
The two players held serve until the seventh game where Gasquet again proved too strong for Youzhny, who is ranked 14 places below the Frenchman.
He went on to take the first set 6-2, but Russian Youzhny responded with a resurgent display in the second, breaking early on to gain control.
Unlike Youzhny, Gasquet was able to find a break back in the sixth game, partly down to a pair of unforced errors from his opponent.
Games were traded until the set reached a tiebreak, which Youzhny led 4-1, but Gasquet fought back, winning six out of the next seven points to claim a straight-sets victory.