Murray made the last-eight for the sixth time with a 7-6 (11/9) 6-4 6-3 win over John Isner of the United States on Sunday.
The 29-year-old, a three-time semi-finalist, will face Gasquet, the last French player standing, in the last-eight. Murray has a 7-3 career lead over the 29-year-old Gasquet, including wins at Roland Garros in 2010 and 2012.
Gasquet outplayed fifth seed Nishikori 6-4 6-2 4-6 6-2 to reach the quarter-finals for the first time.
The ninth seed had trailed Nishikori 2-4 early in the first set, but after an hour-long rain delay, he won nine out of the next 10 games to turn the match on its head.
“I started badly, but everyone helped me and it was difficult for Nishikori to play against that,” he said.