The Swiss ace needed just over an hour and a half to book a fourth-round spot by beating the big-serving German 6-3 7-5 6-2.
The three-time former champion had to take a medical time out just a few games from victory, with the trainer massaging his left thigh.
Nadal will be up against the promising Alex de Minaur, who beat Pierre Hugues Herbert 6-2, 6-7 (8), 7-5, 6-3.
Ken and Neal Skupski, the British doubles pairing, were already two sets up against Aisam-ul-Haq Qureshi and Jean-Julien Rojer when Federer arrived on the court adjacent to their match. "You have to win the last point, and I didn't succeed in that today".
Cilic was runner-up to Roger Federer previous year and showed great form by beating Novak Djokovic in the Queen's Club final last month.
Their defeats means that eight of the women's top 10 at this year's championships have already been knocked out.
Serena took her Wimbledon winning streak to 17 matches, thanks in part to 13 aces.
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Williams and Evgeniya Rodina are the only remaining mothers of the six who had entered this event - and they will face each other in the fourth round Monday. Querrey's career-best finish came at Wimbledon 2017 - where he was a semifinalist - but he won't improve upon that this year.
"Serena is my idol", Rodina said.
Cilic blew a two-set lead in his 3-6, 1-6, 6-4, 7-6 (7/3), 7-5 loss to Argentina's Guido Pella, ranked at 82.
The 23-year-old American threatened a remarkable comeback in the third-round contest, coming from a set and 4-0 down to force a decider.
John Isner, the U.S. ninth seed, saved two match points, fired 64 aces and 88 winners to defeat Belgian qualifier Ruben Bemelmans 6-1, 6-4, 6-7 (6/8), 6-7 (3/7), 7-5.
Zverev erased a 2-1 deficit in sets three times to reach the quarterfinals of the French Open, his best result at a Grand Slam.
A rain delay saw the match delayed overnight with Cilic two sets to the good but he failed to find his rhythm as the Argentine won 3-6 1-6 6-4 6-4 7-5 to reach the third round of a major for the first time.
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