Roddick will face Croatia's Cilic as he bids to reach the semi-finals for the fourth time.
Gasquet, a semifinalist in Miami last month, beat Croatian ninth seed Marin Cilic 7-5 6-4.
CNN: Nadal 'relaxed' as rivals struggle at Monte Carlo Masters
"The conditions were tough, the balls were heavy, " Cilic told the official ATP Tour Web site.
Marin Cilic squared the tie thanks to a 6-4, 7-5, 6-2 win over Viktor Troicki.
Cilic will face Andy Roddick after coming through a marathon fourth-round match after four hours and 38 minutes.
Croatian third seed Marin Cilic cruised into the last eight as he beat Germany's Matthias Bachinger 6-3 6-2.
CNN: Haas rolls back the years to stun top seed Tsonga in Munich
But Murray insisted after his victory over Cilic that regaining the no.2 ranking was not his main priority.
His opponent Wednesday will be No. 9 Marin Cilic, who defeated No. 6 Jo-Wilfried Tsonga 7-5, 7-6 (4).
Cilic dropped the first set but took a 2-1 lead as Del Potro struggled to find his top form.
Fourth seed Murray, who lost in the doubles playing with brother Jamie on Monday, will next face Marin Cilic.
James Blake, Marin Cilic and Gilles Simon have also signed up to appear.
He fought from behind in the quarter finals against Marin Cilic and the semis against Tomas Berdych.
FORBES: Andy Murray's Management Change Pays Off at the U.S. Open
Murray's path to the title was made easier when the Croatian Marin Cilic stunned home favourite Davydenko 1-6 7-5 6-1.
Juan Martin del Potro can wrap up the tie if he beats Cilic in the first singles rubber on Sunday.
He will now play Marin Cilic in the last eight after the Croatian beat Spain's Fernando Verdasco 4-6 6-1 6-4.
The effort seemed to catch up with him against Cilic, and he received lengthy treatment on his left foot during the contest.
Sixth seed Cilic, the winner of ATP titles in Chennai and Zagreb already this year, defeated Arnaud Clement of France 6-2 6-4.
As a result Nalbandian, who had been leading sixth-seeded Cilic of Croatia 7-6 (3), 3-4 when the incident occurred, defaulted the match.
"It's definitely not the way I wanted to win it, " Cilic said.
In Munich, top seed Marin Cilic reached the semifinals of the BMW Open by beating sixth-seeded Spaniard Nicolas Almagro 6-4 6-7 (2-7) 6-2.
Meanwhile, defending champion Marin Cilic will play Swiss third seed Stanislas Wawrinka in the final of the Chennai Open in India on Sunday.
Cilic was joined in round three by French fifth seed Jo-Wilfried Tsonga and Spanish sixth seed Fernando Verdasco, who both recorded straight-sets victories.
Despite a tight opening exchange Cilic went down in straight sets, the second time he has been runner up here in two years.
Cilic, ranked 15th in the world, will next play France's Jeremy Chardy, who beat another German Nicolas Kiefer 6-4 7-6, in Friday's quarterfinals.
And though Cilic twice went a break down in the deciding set, he held on to claim it 6-4 and progress to round three.
In the other men's Wimbledon warmup tournament at London's Queen's Club, Croatian sixth seed Marin Cilic will play veteran Argentine David Nalbandian in Sunday's final.
In other matches, fourth seed David Ferrer beat Ivan Ljubicic 3-6 6-2 6-1, and fifth seed Marin Cilic defeated Janko Tipsarevic 2-6 7-6 (7-3) 6-4.
Del Potro kept unforced errors to 24, less than half Cilic's 53, and converted all of his break points in their quarterfinal match on Thursday.
The Argentine pair triumphed 3-6 7-6 6-3 6-7 8-6 against Marin Cilic and Ivo Karlovic in a match lasting one minute shy of five hours.
应用推荐