She is the fourth former women's champion to get knocked out, joining Maria Sharapova, Petra Kvitova and last year's victor Garbine Muguruza.
The 38-year-old American - last year's runner-up at Wimbledon - was the oldest entrant in the women's singles field.
Nadal won only 64 per cent 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 under 2 1/2 hours on Centre Court.
Rodina, through to the fourth round of a Grand Slam for the first time, was 0-15 against top-20 ranked opponents before Friday's win.
The match on Monday will also be a tussle between two of the tour's mothers. I thought I did very well today.
Roger Federer cruised into the fourth round at Wimbledon with an effortless straight-sets victory against Jan-Lennard Struff.
But third-seeded Cilic unexpectedly fell.
The seven-time champion came from a break down in the first set to get past Kristina Mladenovic 7-5, 7-6 (2). She sat out Wimbledon past year while pregnant.
"Stopping last night was not a nice feeling to go to bed with", said Zverev.
"I knew it was going to be a tough match, so I went onto the court with the determination and intensity, thinking I can't make any mistakes". "I am having a good season - made the quarters at the French Open - it's all coming together".
Cilic last bowed out in the second round in 2013.
His early exit opens up Federer's path to the final even more, with No 8 Kevin Anderson the highest-seeded player left in the Swiss star's half of the draw after No 6 Grigor Dimitrov lost in the first round.