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.
Bertens faces Czech seventh seed Karolina Pliskova or Romanian world number 28 Mihaela Buzarnescu for a place in the quarter-finals.
"The rain coming helped me a lot", the 28-year-old, who had not won a match at Wimbledon in his two previous visits before this year, told BBC Sport.
Serena gave birth to daughter Alexis Olympia in September, while Rodina became a mother to Anna in November 2012.
Cilic was runner-up to Roger Federer a year ago and showed great form by beating Novak Djokovic in the Queen's Club final last month.
Having made 18 unforced errors in each of the first two sets, Giorgi tightened up her game to make just eight in the third and advance to the last 16 at Wimbledon for the first time since 2012.
Defending champion Roger Federer continued his serene progress at a Wimbledon full of shocks with a straightforward victory over Jan-Lennard Struff.
He is on a great run at Wimbledon since the start of 2017, winning 29 consecutive sets here which is his second longest streak after 2005-2006 when he rattled off 34.
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Zverev had trailed two sets to one when his tie with Fritz was halted due to darkness on Thursday. "But I knew if I took my chances today, I could do well".
Ekaterina Makarova of Russian Federation also reached the fourth round, following up her victory over second-seeded Caroline Wozniacki with a 4-6, 6-4, 6-1 win over former semifinalist Lucie Safarova of the Czech Republic.
Nick Kyrgios, seeded 15, unleashed 42 winners to see off Robin Haase of the Netherlands 6-3, 6-4, 7-5.
However, the dominant way he played in his 6-3 7-5 7-5 victory over German Philipp Kohlschreiber suggested he could go further than ever before.