The Spanish fourth seed, fresh from victory in a Madrid Masters triumph that saw him cruise past Serbian nemesis Novak Djokovic in the semis, will meet American Jack Sock in his third-round match later.
With a potential semi-final meeting with Rafael Nadal on the cards, the Serb will be determined to storm past Del Potro, who had forced a decider in their last two meetings this year.
Serbia's Novak Djokovic returns the ball to to Spain's Roberto Bautista Agut during their match at the Italian Open tennis tournament, in Rome, Thursday, May 18, 2017.
However, he regained his composure and found a way through and will now face either Kei Nishikori or Juan Martin del Potro in the quarter-finals, with several high-profile stars already into the last-eight. The 16th seeded Zverev won the first set on a single break of serve and carried the momentum into the second set as he took a 4-0 lead. I can play a bit more aggressive than the way I played tonight, but overall I'm happy.
"I went out there with the belief that if I played well and executed my game plan, I could walk off that court with a win".
In a shock to Maria Sharapova, the French tennis federation refused to grant the five-time Grand Slam victor and twice Roland Garros champion a wild card to either the main draw or qualifying.
The Austrian fought off a double match point at 4-6 in the tiebreaker, then thwarted a third match point at 6-7 before surviving.
Isner next faces another big server in sixth-seeded Marin Cilic, who dismissed ninth-seeded David Goffin 6-3, 6-4.
If one looks at the head-to-head record, Djokovic has been mighty impressive, winning 13 matches and only losing four.
In the women's draw, Venus Williams beat Johanna Konta 6-1, 3-6, 6-1.
Estonian qualifier Anett Kontaveit, who beat top-ranked Angelique Kerber in the second round, continued her dream debut.
Alexander Zverev, the youngest player ranked in the ATP's top 50, emphasised his burgeoning credentials on clay ahead of Roland Garros later this month by seeing off the challenge of Italian home favourite Fabio Fognini 6-3 6-3 at Foro Italico.
Zverev, the 16th seed who is the youngest player in the ATP top 50, will meet the victor of the clash between Tomas Berdych and Milos Raonic, to be played later Thursday.