Agassi was joined in the last 16 of the tournament by fourth seed Tim Henman, who made light work of Chile's Nicolas Massu 6-3 6-2, and Argentine Guillermo Coria, who disposed of Nicolas Kiefer 6-1 6-1.
Top seed Andy Roddick demonstrated his growing maturity on clay by beating former French Open champion Alberto Costa 6-4 7-5.
Though ranked nearly 50 places below the American, Costa knocked world number one Roger Federer out at the same stage of last year's tournament.
"He's been one of the best claycourt players over the last 10 years so this is very good for my confidence," said Roddick, who has often struggled to adapt his game to the slow surface.
Having fended off an early break point, Roddick settled into the match and took the first set with a combination of heavy groundstrokes and intelligent net play.
The second set followed the same pattern, with Roddick rallying to break Costa and secure the victory in the 12th game.
Qualifier Gasquet ended Federer's 25-match winning streak at the Monte Carlo Masters last month but did not show the same killer instinct against the 35-year-old Agassi.
Early unforced errors meant Agassi was 40-0 down in his first service game but Gasquet's failure to break serve was punished in the next when he lashed a backhand into the net with break point against him.
Two games later, the 18-year-old double-faulted on break point to effectively hand Agassi the set.
Gasquet's nerves got the better of him in the second set too, when he netted a pair of routine groundshots to drop serve in the fifth game. He then double-faulted again in the ninth with match-point against him.
Second seed Marat Safin takes on Spanish qualifier Nicolas Almagro later on Wednesday.
