World number one and four-time winner Djokovic defeated Poland's Hubert Hurkacz 7-5, 6-7 (5/7), 6-1, 6-4.
He will next face either Canada's Felix Auger-Aliassime who, at 18, is the youngest man in the draw, or Ugo Humbert of France, the world number 66.
"He was fighting. He was playing well, serving well and hitting clean and accurate shots," said Djokovic who is now level with Becker in third place on the all-time list for last-16 places.
Roger Federer and Jimmy Connors lead the way with 16.
"It was a great fight the first couple of sets. He deserved to win the second one."
However, Kevin Anderson, the 2018 runner-up, was dumped out, leaving just four of the top 10 seeds in the tournament.
Anderson slumped to a shock third round 6-4, 6-3, 7-6 (7/4) defeat to Argentina's Guido Pella, the 26th seed.
The South African's defeat means that only Djokovic, Roger Federer, Rafael Nadal and Kei Nishikori remain of the top 10 players who started out.
"He made life really difficult for me," said Anderson, playing just his second tournament since March after recovering from an elbow injury.
Pella goes on to face 2016 runner-up Milos Raonic for a place in the quarter-finals.
"I played an unbelievable three sets," said Pella, who had never gone beyond the third round of a Grand Slam tournament in seven years.
In the women's event, Czech third seed Karolina Pliskova reached the fourth round for the second successive year with a 6-3, 2-6, 6-4 win over Hsieh Su-wei of Taiwan.
Pliskova, who fired 14 aces and 42 winners, will next face compatriot Karolina Muchova, the world number 68, who put out Estonian 20th seed Anett Kontaveit 7-6 (9/7), 6-3.
Muchova is making her Wimbledon debut as is 19-year-old Dayana Yastremska and the 35th-ranked Ukrainian marked the occasion by also making the last 16.
She put out Switzerland's Viktorija Golubic 7-5, 6-3 and next meets unseeded Chinese player Zhang Shuai.
World number 60 Zhang made the fourth round for the first time, coming back from 0-4 down in the opening set to beat former world number one Caroline Wozniacki 6-4, 6-2 who failed to see eye-to-eye with the Hawk-Eye line call review system.
Before this year, the 30-year-old Zhang had never won a main draw match at Wimbledon in five attempts.
It was a record which reflected her previous struggles -- she was 0-14 at the Slams until she broke through to the 2016 Australian Open quarter-finals.
On Friday, she fired 26 winners against just eight for Wozniacki, the 2018 Australian Open champion.
- 'I don't have a next match' -
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Wozniacki's affection for the tournament would not have been helped by her unhappiness at what she claimed to be a number of poor calls by Hawk-Eye.
"You trust that it tells you the right thing. Sometimes you do see the balls a little differently than Hawk-Eye," said the 28-year-old.
The umpire, she claimed, offered to get the system looked at -- for the next match on Court Two.
"Well, I don't have a next match," she grumbled.
Zhang, the first Chinese woman in the last 16 in five years, said: "These three matches, I've moved well. If I want to win, I must be focused, clear my mind and play my game."
Raonic, the 15th seed, reached the fourth round for the fifth time by seeing off the sport's tallest man Reilly Opelka of the United States 7-6 (7/1), 6-2, 6-1.
France's Benoit Paire booked a last 16 place for the second time with a 5-7, 7-6 (7/5), 6-3, 7-6 (7/2) win over Czech qualifier Jiri Vesely.
He will meet Spain's Roberto Bautista Agut who stunned Russian 10th seed Karen Khachanov 6-3, 7-6 (7/3), 6-1.
Romanian seventh seed Simona Halep came from 1-3 down in the first set to defeat fellow former world number one Victoria Azarenka 6-3, 6-1.
Azarenka was undone by 33 unforced errors.
Gauff, the 15-year-old sensation who defeated five-time champion Venus Williams in the first round, takes on Slovenia's Polona Hercog for a place in the last 16.
Gauff, ranked at 313, is the youngest player to get to the third round since Jennifer Capriati in 1991 and came through qualifying.
Should Gauff win she'll face Halep for a quarter-final spot.
Andy Murray, making his Grand Slam comeback after hip surgery, was due to team up with Serena Williams in the mixed doubles but with singles matches ending late, their debut was postponed for the day.