Djokovic, the three-time champion, needed treatment on his left thigh on Court 2 but still had too many weapons for Argentina's Horacio Zeballos, winning 6-1, 6-2, 6-3.
"A lot of times you just have to weather the storm and play better on those big points".
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.
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.
Madison Keys became the seventh woman seeded in the top 10 to crash out of Wimbledon as the American was stunned by Russian qualifier Evgeniya Rodina on Friday.
Federer, the only semi-finalist from a year ago left in either of the singles draws, will face France's Adrian Mannarino on Monday.
"Serena is my idol", Rodina said. Williams beat French tennis player 7-5 7-6 (7/2) in the third round on Friday, keeping alive her hopes of a 24th Grand Slam win.
The American player won five games in a row to level and then broke back to make it 4-4 in the third set, but Rodina earned another break to secure victory.
Fifth seed Juan Martin del Potro swept past Feliciano Lopez 6-4 6-1 6-2 in less than two hours, teeing up a clash with France's Benoit Paire, who beat 26th seed Denis Shapovalov 0-6 6-2 6-4 7-6 (7/3).
The 17-times Grand Slam champion dug in though and managed to avoid being dragged further into the afternoon, claiming victory when Kukushkin's forehand smacked into the net tape.
The 52nd-ranked Giorgi produced a backhand victor at 4-5, 30-40 in the second set, then won a tiebreaker to force a deciding set.
Makarova, who has reached the round of 16 at the All England Club for the third time, will next face Italy's Camila Giorgi, who advanced after saving a match point while beating Katerina Siniakova of the Czech Republic 3-6, 7-6 (6), 6-2.
In the men's tournament, fourth-seeded Alexander Zverev completed another excellent comeback to reach the third round.