The 36-year-old American will face another mother in the fourth round Monday: 120th-ranked qualifier Evgeniya Rodina of Russian Federation, who upset No. 10 seed Madison Keys of the U.S.
Cilic looked headed for a routine victory before the match was suspended by rain in the third set on Wednesday (Thursday NZT) but the former US Open champion couldn't find the same rhythm.
But, on the resumption, Pella improved and wrapped up the third set and took the fourth on a tie-break.
"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". "You have to win the last point, and I didn't succeed in that today".
THe 32-year-old took 10 minutes to hold his opening service game and was still being asked questions by the world number 77 in the third when he slipped 3-1 behind in the face of some inspired shot-making by his opponent.
The Kazakhstani was made to pay for converting just two of 13 break points in the match. "Probably I should go to toilet".
Fritz was leading 4-6, 7-5, 7-6 (0).
Stan Wawrinka also exited, losing to Italian qualifier Thomas Fabbiano 7-6 (7) 6-3 7-6 (6).
He will face Australian teenager Alex De Minaur in the next round.
Seven-time champion Serena Williams reached the Wimbledon fourth round on Friday with a 7-5, 7-6 (7/2) win over France's Kristina Mladenovic.
Barty came back from 2-5 down in the second set to see off 2014 runner-up Bouchard who came through qualifying this year.
For more sports coverage, visit the ABS-CBN Sports website.
Saisai ended a run of three successive first round losses at the All England Club when she beat compatriot Qiang Wang on Tuesday.