Cilic looked headed for a routine victory before the match was suspended by rain in the third set on Wednesday, but the former US Open champion couldn't find the same rhythm on Thursday.
"Yesterday he was playing so, so good, and hitting the ball so hard that I couldn't do anything," the 82nd-ranked Pella said. "So the rain helped me a lot."
It's the first time Pella has reached the third round of a Grand Slam.
Djokovic called for a trainer and had his left leg massaged when leading 4-3 in the third set. But he easily held serve in the next game and broke Zeballos again to close out the match.
"I've been doing checkups now," Djokovic said. "It seems like it's nothing major. Tomorrow I'll see on the practice session how it feels. Hopefully it's going to be fine."
Nadal won only 64 percent of points on his first serve and faced 13 break points -- including three in the first game of the match -- but he saved 11 of them and converted five of his own to wrap up the win in 2 hours, 23 minutes on Centre Court.
No. 8 Kevin Anderson beat Andreas Seppi on Thursday to set up a third-round meeting with Philipp Kohlschreiber of Germany. The hard-serving Anderson hit 34 aces against Seppi to win 6-3, 6-7 (5), 6-3, 6-4.
The 25th-seeded Kohlschreiber ousted Gilles Muller of Luxembourg 7-6 (6), 7-6 (4), 7-6 (3) in a match that featured no breaks of serve.
Meanwhile, ninth-seeded John Isner hit 64 aces and saved two match points to reach the third round of Wimbledon for the fourth time. The American beat Ruben Bemelmans 6-1, 6-4, 6-7 (6), 6-7 (3), 7-5 in a match that had been suspended by rain on Wednesday.
Isner's ace total is the third-highest ever at Wimbledon -- behind his own record in a three-day epic against Nicolas Mahut in 2010, which went to 70-68 in the fifth and is the longest tennis match in history. Isner had 113 aces in that match, and Mahut hit 103.
It was Isner's first win in a five-setter at Wimbledon since the Mahut match. He has never reached the fourth round at the Grand Slam tournament.
Benoit Paire of France ousted 26th-seeded Denis Shapovalov 0-6, 6-2, 6-4, 7-6 (3). The 47th-ranked Paire will next face fifth-seeded Juan Martin del Potro after the Argentine defeated Feliciano Lopez 6-4, 6-1, 6-2.
Having raced through the first set in just 19 minutes -- making only one unforced error -- 26th-seeded Shapovalov was unable to maintain his level.
Paire took advantage, with the Frenchman losing just seven of 60 points on his first serve from the end of the first set onward.
But Stan Wawrinka couldn't muster another comeback at Wimbledon, losing to Italian qualifier Thomas Fabbiano 7-6 (7), 6-3, 7-6 (6) in the second round.
Information from The Associated Press was used in this report.