Rain interrupted the fourth-round match between Murray, the No. 4 seed, and the 16th-seeded Cilic Monday and disrupted play again Tuesday.
"We stopped, obviously, a lot," said Murray, who has reached the semifinals each of the past three years. "You're always a little bit edgy when you come out after a rain delay when you have the momentum with you. You stop, and then you come back out, and it feels a bit like you start from scratch again. I needed to serve well today and I got a lot of free points that helped me out."
Murray will play Ferrer, the No. 7 seed, next. Ferrer defeated ninth-seeded Juan Martin del Potro 6-3, 6-2, 6-3 under the roof on Center Court.
Tsonga, seeded fifth, waited out a lengthy rain delay before eliminating 10th-seeded American Mardy Fish 4-6, 7-6 (7-4), 6-4, 6-4 in 3 hours, 18 minutes on Court 2. Tsonga produced 20 of the match's 37 aces and he broke Fish's serve three times.
Tsonga took a break after the second set to be treated for a sore back.
Also advancing Tuesday were 27th-seeded German Philipp Kohlschreiber and his 31st-seeded countryman Florian Mayer. Kohlschreiber took care of American Brian Baker 6-1, 7-6 (7-4), 6-3. Mayer eliminated 18th-seeded Frenchman Richard Gasquet 6-3, 6-1, 3-6, 6-2.
Kohlschreiber will meet Tsonga, while Mayer goes against top-seeded Novak Djokovic, the defending champion who eased past fellow Serb Viktor Troicki Monday, 6-3, 6-1, 6-3.
Also winning Monday were third-seeded, six-time champion Roger Federer and No. 26 seed Mikhail Youzhny. Federer ousted Belgian Xavier Malisse and will next take on Youzhny, who took five sets to beat Uzbekistan's Denis Istomin.
Federer, who reached seven straight Wimbledon finals from 2003-09, is gunning for his first Grand Slam title since the 2010 Australian Open.