Bertens faces Czech seventh seed Karolina Pliskova or Romanian world number 28 Mihaela Buzarnescu for a place in the quarter-finals.
Serena will play world number 120 Rodina if she advances to the fourth round.
The 38-year-old American - last year's runner-up at Wimbledon - was the oldest entrant in the women's singles field.
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. She missed last year's tournament while pregnant.
"Serena is my idol", Rodina said.
Keys lost nine games on the trot from 5-2 up in the first set to 0-4 down in the second.
No. 8 Kevin Anderson beat No. 25 Philipp Kohlschreiber of Germany 6-3, 7-5, 7-5 after losing just eight of 61 points on his first serve, while Gael Monfils of France advanced to the fourth round for the first time by knocking out last year's semifinalist Sam Querrey 5-7, 6-4, 6-4, 6-2. She then broke back to level the third set at 4-4, but Rodina immediately earned another break and served out the victory.
Horner: Max Verstappen hasn't changed his style
Verstappen's Red Bull team-mate Australian Daniel Ricciardo was also forced to retire on his 29th birthday. Horner said: "The whole driver market is waiting for Toto's driver to kick that off".
Federer, making his 20th consecutive appearance at the grass-court Grand Slam, is yet to face a break point from his opening three matches and will now continue his campaign against 22nd-seed Adrian Mannarino. After the double fault in the first set, Mladenovic appeared to lose her confidence.
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.
"I tried to hit the ball hard and fight for every ball and I think in the end that's why I won".
The Swiss defeated the Croat in the 2017 final at the All England Club, while he was pushed to five sets in the final of the Australian Open earlier this year.
He will next face Mannarino after the French player held off a fightback from Daniil Medvedev to progress 6-4 6-3 4-6 5-7 6-3.