Nadal, meanwhile, was similarly impressive, dispatching Czech Republican star Jiri Vesely to win 6-3 6-3 6-4.
Roger Federer is surprising even himself with his latest record-setting run to the Wimbledon quarter-finals.
Today it was the first match he faced break points since last year's final against the Croatian Marin Cilic.
"It was crucial for him to stay in the match at the beginning of the second set and then it got tougher", said 20-time major victor Federer.
"I can be here, I can't come here and say, No, of course I want to play against the best players".
Anderson is the first South African man in the Wimbledon last-eight since Wayne Ferreira in 1994.
Federer was first on Centre Court on Monday for what is known as Wimbledon's "Manic Monday", in which all 32 remaining singles players vie for a place in the quarterfinals.
There are uncertainties in that lineup - but the story at the end of the drama is likely to be the same as it has been for the past seven grand slam tournaments: victory for either Federer or Nadal.
"It was a really high quality match, I had to dig deep", Anderson said after the match.
I deserved penalty for Hamilton clash - Raikkonen
Ferrari lead the constructor standings with 287, ahead of Mercedes with 267. Ferrari's Sebastian Vettel celebrates winning the race.
When asked whether he would rather not play Federer in the final, should they meet, Nadal said, "If he's in the final, I am excited to play Roger".
The world No.1 and reigning French Open champion also extended his grand slam streak to 11 matches to book an appointment on Thursday with either Argentine fifth seed Juan Martin del Potro or veteran Frenchman Gilles Simon.
Del Potro was leading 7-6 (7/1), 7-6 (7/5), 5-7 when his tie with Simon was halted for the night due to darkness.
Djokovic, seeded 12th, won easily over Russia's Karen Khachanov, 6-4, 6-2, 6-2.
"He's got weapons, a big serve and a big forehand", said Djokovic of the world number 40. I feel like I'm playing some of the best tennis of my career. I served well when I needed to. Anderson admitted his triumph against Monfils, which lasted over 3½ hours, was demanding, and as a result he will train lightly Tuesday, focusing more on recovery and plotting his game plan to tackle Federer. "I think I can be very pleased".
The victor of that clash will take on either Japan's 24th seed Kei Nishikori or unseeded Ernests Gulbis.
And now, heading into the quarterfinals, Federer hasn't had to waste time playing needless sets. When I'm doing that, I think I'm a very unsafe player.
Later Monday, two-time champion Rafael Nadal attempts to make the quarter-finals for the first time since he finished runner-up in 2011. She won the Australian Open and US Open in 2016 - becoming the first German to win a Slam since Steffi Graf in 1999 - and reached the Wimbledon final that year.
Federer admitted he was surprised by the quick nature of the first set, which kick-started his progress to a 16th Wimbledon quarter-final.
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