Murray next faces the victor of the match between American John Isner and Russian Karen Khachanov which will be played at a later time.
But it was Del Potro who made the first breakthrough, with his powerful forehand helping him into a 3-1 lead.
"It was obviously an important win for me and a big match, because Juan, when he's playing well, is one of the best players in the world", Murray said. Del Potro had four set points in total and double-faulted at 7-6 up in the tie-breaker.
Murray, Del Potro set for new epic?
The 33-year-old Verdasco converted eight of 13 break points punishing Pablo Cuevas, 6-2, 6-1, 6-3.
Del Potro, the 2009 U.S. Open champion, was playing at his first French Open since 2012 because of a series of operations on his left wrist.
He retrieved it when Murray served for the set in the 10th game before handing the advantage straight back.
Murray seized his lifeline, a 4th ace giving him a two-set lead.
Murray, who was joined by 2015 French Open champion Stan Wawrinka in the fourth round, did enough to keep his nose in front throughout the encounter, while del Potro was unable to reprise the heroics that saw the pair produce such magic in last year's Rio Olympics gold medal match.
And Petchey can see the world No 1 improving as he is progressing at the tournament. "He is one of the smartest guys on the circuit".
Deaf singer wows crowd, judges on 'America's Got Talent'
Mandy been singing since she was 4, and deaf since she was 18. "But I want to do more with my life than just give up". Using "muscle memory, visual tuners, and trusting my pitch", Harvey said she found a way to keep up with her passion.
Elsewhere, Marin Cilic, a former US Open champion, defeated an ailing Feliciano Lopez 6-1, 6-3, 6-3.
Nishikori beat Murray in a dramatic five-setter 1-6, 6-4, 4-6, 6-1, 7-5 a year ago to reach the semifinals at Flushing Meadows.
Simona Halep showed that an ankle injury is no longer a huge worry as she beat Russian Daria Kasatkina 6-0, 7-5.
Murray will meet either eighth seed Kei Nishikori in the quarter-finals.
It was the first time in the Open era that two Asian men had met as late as the third round in a Grand Slam tournament.
Not on speaking terms over a Fed Cup-related controversy, Caroline Garcia (pic) and Alize Cornet take their differences to court in the fourth round on Philippe Chatrier. I was really looking forward to the French Open because I have struggled over the last six or seven weeks coming in.
Cornet punched the air with delight after sealing victory on her next chance with a clipped backhand at the net.
"I expected a very tough match".
"It's not quite Wimbledon, when I get to stay in my own bed every night, but it feels a little bit more like normal home life than the other slams".
Fifth-seeded Elina Svitolina advanced to the French Open quarterfinals by beating Croatian qualifier Petra Martic 4-6, 6-3, 7-5.
Spanish No. 21 seeded Carla Suarez defeated Ukrainian 14th seed Elena Vesnina 6-4, 6-4 in an hour and 43 minutes.