The number two seed Briton played with power and control early on, breaking when the ninth-seeded Frenchman double-faulted to lose the second game and consolidating to lead 5-2.
But overuse of the drop shot cost the Scot as Gasquet, the last French player in the singles draw, reeled off five games in a row to take the first set.
Murray also led 5-2 in the second set before Gasquet rallied again, forcing a tiebreak that Murray won 7-3 before the Briton - reading Gasquet's serve with more confidence and bringing his superior fitness to bear - took sets three and four with the loss of just two more games.
He will next face Wawrinka after the Swiss third seed beat Spain's Albert Ramos Vinolas in straight sets.
Wawrinka held off a late charge by Ramos Vinolas to enter the semi-finals with a 6-2 6-1 7-6(7) victory on Wednesday.
The third-seeded Swiss, who got off to a slow start in opening rounds of the claycourt grand slam, steamrollered through the first two sets and resisted in the third on Court Suzanne Lenglen.
Wawrinka used his single-handed backhand to move Ramos-Vinolas left, right and centre and it took the unseeded left-hander more than two sets to adjust.
After breaking back in the third, he forced a tiebreak and even had a set point but Wawrinka saved it and raised his arms in celebration when Ramos Vinolas's forehand sailed long on match point.
"By the end Ramos played better and I made mistakes. But I'm very satisfied that I won this in three sets," he said courtside.