It so easily could have been eight, with Dzumhur leading by a break on two occasions in the critical fourth set and serving for the match at 6-5 up, while also crafting a match point on the Zverev serve deep into the decider.
Up next for Zverev, who has never progressed beyond the fourth round of a Grand Slam, is the victor of Lucas Pouille's match with Karen Khachanov.
It is a second five-set win in a row for the 21-year-old.
Sensing an opportunity Zverev stormed into a 5-1 lead in the tiebreak before levelling the match.
"It was the first time I have won on this court and I hope there are many more to come".
Djokovic has now made the last 16 of a Slam for the 43rd time, tying Jimmy Connors on the all-time list, second only to Roger Federer´s tally of 60.
The match didn't end well for the Bosnian either.
La Belgique pleure les victimes de l'attaque revendiquée par l'EI — Liège
Les jours de l'un d'entre eux sont en danger " puisque l'artère fémorale a été touchée", a précisé Christian Beaupère. Deux policiers ont été blessés, a de son côté affirmé une source policière. "Il n'y a pas d'autres mots".
Meanwhile, fourth seed Grigor Dimitrov fell 7-6 (7-4) 6-2 6-4 to the experienced Fernando Verdasco, with the Bulgarian exiting at the third-round stage for a third time on what is undoubtedly his weakest surface.
"None. Mainly I was thinking what I was going to have for lunch", said a deadpan Zverev after completing his 6-2 3-6 4-6 7-6 (3) 7-5 great escape.
He's also somehow only made the fourth round once before, at Wimbledon a year ago. "It was a very tight match".
The Australian Open quarter-finalist had never reached the second week in Paris in seven previous attempts, with Verdasco's 7-6 (7-4) 6-2 6-4 win extending that streak. "It´s been proven to work in the past".
Anderson will face Argentina's world No 12 Diego Schwartzman for a place in the quarter-finals on Monday.
When Jonathan tried to turn the subject back to sport, telling the rising star "you clearly love your tennis", Zverev responded 'not as much as your accent!'