Eight-time champion Federer will be playing in his 53rd Grand Slam last-eight when he tackles either Gael Monfils of France or Kevin Anderson, the eighth-seeded South African.
Top seed Federer, 36, has now won 32 consecutive sets at Wimbledon, just two behind his record set from the third round in 2005 to his title triumph in 2006.
"It was crucial for him to stay in the match at the beginning of the second set and then it got tougher," said Federer after his sixth win in six meetings against French left-hander Mannarino.
"You always tend to play better against better players and I am happy to be back in the second week of Wimbledon."
Federer boasts a 4-0 career record against 2017 US Open runner-up Anderson and 9-4 over Monfils.
Monfils and Anderson have yet to get to the quarter-finals at Wimbledon, although the French shotmaker may just be the slight favourite as he has a 5-0 winning record over the big South African.
Later on Monday, two-time champion Rafael Nadal attempts to make the quarter-finals for the first time since he finished runner-up in 2011.
The world number one faces Czech world number 93 Jiri Vesely, who has never got past the fourth round of a Grand Slam.
Victory for Nadal over his fellow left-hander would take him into a 35th Grand Slam quarter-final.
It would also give him a last-eight clash against Juan Martin del Potro, the fifth seeded Argentine, or unseeded Gilles Simon of France.