The 21-year-old German looked to be on the brink of another disappointing early exit from a Grand Slam, with 26th seed Dzumhur serving for the match at 5-4 in the fourth set.
But Zverev held his nerve to force a tiebreak, which he won comfortably and eventually prevailed in the final set to book his first trip to the last 16 of the tournament.
Former champion Novak Djokovic battled with Spaniard Robert Bautista Agut to grind out a 6-4 6-7(6) 7-6(4) 6-2 victory.
The 20th-seeded Serb is playing at his lowest seeding in a Grand Slam since 2006 having dropped to 22 in the world rankings but has now reached the last 16 of a major for the 43rd time -- second on the all-time list behind Roger Federer.
Fourth seed Grigor Dimitrov struggled to discover his touch and lost 7-6(4) 6-2 6-4 to Spanish claycourt specialist Fernando Verdasco.
The Bulgarian fell to his seventh consecutive defeat against top 50 opponents at the French Open.
In the women's draw, fourth seed Elina Svitolina's French Open breakthrough was again put on hold as the Ukrainian fourth seed suffered a 6-3 7-5 loss in the third round to Romania's Mihaela Buzarnescu.
Earlier on Friday, US Open runner-up Madison Keys saved two set points in the second set before putting away Japan's Naomi Osaka 6-1 7-6(7) to reach the fourth round.
