The Serb next faces home hope Kyle Edmund, a 6-4 7-6 (7-0) 6-2 victor over American Bradley Klahn.
But, on the resumption, Pella improved and wrapped up the third set and took the fourth on a tie-break.
While the result was bad news for Cilic, Federer must have been watching with a wry smile as he avoided the Queen's champion.
"I felt much more confident in my game, I served better and I felt so pumped", said Pella, who reached the third round of a major for the first time. "So the rain helped me a lot".
"Today, I started to play more aggressively and fight for every ball and that is why I won". I think I was very quick during the match.
Torreira ready for 'scary' Arsenal move
Torreira will hope to have another big impact for Uruguay when they take on France in their World Cup quarter-final on Friday. Lucas Torreira has turned down the opportunity to undergo a medical in Russian Federation ahead of his move to Arsenal .
Unfortunately, in the important moments the world's number 77 player couldn't grasp the nettle.
Nick Kyrgios eased past Robin Haase 6-3 6-4 7-5, to book a third-round clash with Kei Nishikori, who accounted for Bernard Tomic 2-6 6-3 7-6 (9/7) 7-5.
"I didn't know but I don't see a clock here".
South Africa's Kevin Anderson, seeded eight, swept past Germany's Philipp Kohlschreiber 6-3, 7-5, 7-5 on the back of 22 aces and 42 winners.
Ninth-seeded John Isner hit 64 aces and saved two match points to beat Ruben Bemelmans 6-1 6-4 6-7 (6) 6-7 (3) 7-5 in a match that was also suspended overnight.
Highly-rated American Frances Tiafoe edged out France's Julien Benneteau 4-6 6-3 6-4 6-2.
Second seed Nadal is in to the third round for the eighth time where he will face Australian teenager Alex De Minaur. "It's more hard for me to understand what's going on", said the 32-year-old Nadal after his 6-4, 6-3, 6-4 win.
German fourth seed Alexander Zverev takes on Taylor Fritz, the world number 68, who is bidding to reach the third round of a Slam for the first time.