Second seed Federer clinched a 6-4, 6-4, 6-7 (5/7), 6-2 victory over world number 69 Groth who fired a 147mph ace in the third game.
Federer will next face Spanish 20th seed Roberto Bautista Agut for a place in the quarter-finals.
"I am very happy. It has been a hot week the first week but thankfully I have had easy matches going through without too many long four or five setters," said Federer.
Dreadlocked German qualifier Brown was unable to follow his triumph over Nadal when he slumped to a 6-4, 7-6 (7/3), 4-6, 6-3 defeat to Serbian 22nd seed Viktor Troicki.
Brown joins Lukas Rosol, Steve Darcis and Nick Kyrgios who all lost their next match at Wimbledon after knocking out Nadal over the last four years.
Troicki goes on to face either British wildcard James Ward or Canada's Vasek Pospisil for a place in the quarter-finals.
It is the 29-year-old Serb's first last-16 run at Wimbledon since 2012 and his best performance at a Grand Slam since returning from a 12-month drugs ban in July last year.
"The schedule is to go to the airport and take the earliest flight, because tomorrow morning, I have club matches in Cologne, Germany," said the 30-year-old.
US Open champion Marin Cilic took just two games to see off John Isner on Saturday after returning to finish off their third round match poised at 10-10 in the final set.
- First timers -
Croatian Cilic, the ninth seed, beat the US 17th seed 7-6 (7/4), 6-7 (6/8), 6-4, 6-7 (4/7), 12-10 on Court One in a match that lasted four hours and 31 minutes in total after it had been suspended late Friday due to bad light.
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