In the men's tournament, Roger Federer's latest victory never looked in doubt as he won 37 of 40 first-serve points and stretched his Wimbledon winning streak to 29 sets, beating Jan-Lennard Struff of Germany 6-3, 7-5, 6-2. "I think I played much better today", Nadal said.
The 36-year-old American now can not meet a top 32 player until the semi-finals after the scattering of the seeds this week.
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.
Anderson earlier became the first man through to the fourth round after beating Philipp Kohlschreiber 6-3 7-5 7-5.
"It was very tough because he played very well", said 11-time French Open champion Nadal, who is chasing an 18th major.
Venus, seeded ninth, joined Petra Kvitova, Maria Sharapova, Caroline Wozniacki and Sloane Stephens as the most high-profile casualties of Wimbledon's year of the giant-killers in the women's singles.More news: World Cup big guns primed for last-eight battles
Fourth-seeded Alexander Zverev reached the third round in the men's event by beating Taylor Fritz of the US 6-4, 5-7, 6-7 (0), 6-1, 6-2 in a match that was suspended a day earlier after the third set.
"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.
Defending champion and top seed Federer, by contrast, has made the third round without breaking sweat. "But I knew if I took my chances today, I could do well".
Rodina, a 120th-ranked Russian qualifier, upset Keys 7-5, 5-7, 6-4.
However, she lost her way after that and went a set and 4-0 down before recomposing herself and taking it to a third.
"I worked really, really hard (to get back playing) and it's been a long, arduous road but I always expected to come out and do the best I can do".
Meanwhile in the doubles, South Africa's Raven Klaasen and his New Zealand partner Michael Venus advanced to the second round when they defeated Australia duo Lleyton Hewitt and Alex Bolt 7-5, 6-2, 3-6, 6-1.