The Serbian world number one hit back strongly after losing a tight opening set on a tiebreak but trailed in both the third and fourth sets before turning them around.
Germany’s Zverev will rue allowing Djokovic back from 4-1 down in the third set and he was also up 3-0 in the fourth.
Djokovic again wore tape on his abdominal area, a legacy of the injury picked up in his third-round win over Taylor Fritz. But it did not appear to bother him as he reached his 39th Grand Slam semi-final, where he will face an unlikely opponent in Russian qualifier Aslan Karatsev.
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The 23-year-old Zverev began in confident fashion and secured the first break of the match but failed to convert a set point on Djokovic’s serve when leading 5-3.
He wavered on his own serve at 5-4 though, and Djokovic took the set into a tiebreak, only for some uncharacteristic errors to allow Zverev to sneak a 59-minute set.
Djokovic needed only 29 minutes to win the second set but his level dropped at the start of the third, and he destroyed a racket in anger as he slipped 4-1 behind.
Zverev was 0-30 ahead on Djokovic’s serve in the next game but Djokovic found another gear, winning 20 of the next 24 points to move one set from victory.
Another momentum shift saw Zverev take a 3-0 lead in the fourth set and have three points to move 4-0 ahead, but Djokovic dug in to haul himself back into contention.
Zverev threatened to take the match into a decider but Djokovic saved a set point at 5-6 with an ace and a powerful first serve brought up match point in the day’s second tiebreak.
Djokovic netted a backhand but forced a volley error from Zverev to earn another chance, this time on his own serve, and he wrapped up the match with his 23rd ace.