The top seed came from behind to win 6-7 (6/8), 6-7 (6/8), 6-1, 6-4, 7-5 in a last-16 tie suspended due to bad light on Monday at the end of the fourth set.
The Serb, who has now reached 25 Grand Slam quarter-finals in a row, goes on to face Croatia's US Open champion Marin Cilic for a place in the semi-finals.
Djokovic will be the overwhelming favourite for that match on Wednesday -- he leads their head-to-head 12-0.
Anderson, the South African 14th seed, fired 40 aces and 71 winners but committed 41 unforced errors.
"I think Kevin served exceptionally well. I found it one of the most difficult matches I've played in my Wimbledon career. At times I was really helpless with my returns. He was very aggressive," said Djokovic.
"I thought it was high-quality tennis in the fifth set.
"I was two sets down. To come back and win in five gives me great satisfaction and confidence for the next challenge."
After their match was suspended due to fading light on Monday, it was the 29-year-old Anderson who was the more aggressive first up Tuesday.
But Anderson eventually paid a heavy price for his all-or-nothing approach as he tried to break through into a Grand Slam quarter-final for the first time in 26 attempts.
He served up two double faults to go match points down in the 12th game and Djokovic secured victory when the South African bungled a volley off a blistering service return by the top seed.
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