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.
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.
"He re-started badly (after the rain) and that gave me confidence to play my game and go for my shots."
The last French winner of the men's singles title at Roland Garros was Yannick Noah in 1983.
Murray will be playing in his 20th quarter-final at the last 21 majors after braving an Isner storm in the first set where he faced down three set points in the tie-breaker.
Isner was bidding to become the first American man in the last-eight in Paris since Andre Agassi in 2003.
- Rain delay -
Having not allowed Murray a single break point, the 31-year-old squandered the three set points before the British star pounced to pocket the opener.
Rain forced the pair off Suzanne Lenglen court for an hour with Murray 2-1 ahead in the second set before Isner was broken for the first time in the 10th game.
Murray broke for 3-1 in the third while Isner clung on saving two more break points in the sixth game.
But the world number two wrapped up victory -- and his sixth win in six clashes with the big American -- with his ninth ace of the tie.
Isner was undone by Murray's superior returning which contributed to his 57 unforced errors.
Murray said he had wanted the match halted earlier as the rain made conditions dangerous.
"I know it's difficult sometimes when the right time to stop is, but I think on clay courts that the players really need to be the ones that kind of decide that," he said.
"If they don't feel comfortable then you have to stop, because it's a surface if you get that wrong you can hurt yourself."
Wawrinka, the third seed, saw off Serbia's Viktor Troicki 7-6 (7/5), 6-7 (7/9), 6-3, 6-2 for his 11th straight win in the year's second Grand Slam.
He will next face a Spanish left-hander who isn't named Rafael Nadal after unheralded Albert Ramos-Vinolas reached his first Grand Slam last-eight with a 6-2, 6-4, 6-4 defeat of Canadian eighth seed Milos Raonic.
Sunday was a fifth win in five meetings for Wawrinka over Serbia's Troicki whose challenge fizzled out following a right thigh injury at 4-1 down in the third set.
"It was far from easy with tough conditions -- heavy and cold, but I am happy to have come through it," 31-year-old Wawrinka said.
- Shock winner -
Ramos-Vinolas, 28, had never got beyond the second round of any major before this Roland Garros and had failed to win a match at the tournament since 2011.
But the world number 55, who is only his country's ninth best player, ensured a left-hander from Spain would be in the last-eight after the injury-enforced withdrawal of nine-time champion Nadal.
"I had lost four times in a row here so I am very happy," said the shock winner who had also won just four matches in his entire career at the majors before coming to Paris.
"I played a great match. I think the cloudy conditions helped me as they made the court slower."
He trails Wawrinka 6-0 in career meetings including last week in Geneva where he won just two games.
"Wawrinka is the titleholder. He's hugely powerful. Last week I played him, and he won very easily," said Ramos-Vinolas whose only other previous win over a top 10 player was against Roger Federer in Shanghai in 2015.
source: www.abs-cbnnews.com
