Rotterdam: Del Potro, the number one seed, moved safely through with a victory over unseeded Frenchman Paul-Henri Mathieu 6-2, 7-6 (7/2) in 86 minutes, his win featuring 10 aces.
Meanwhile, second-seeded wild card Murray had a tougher time before eventually overcoming promising Austrian qualifier Dominic Thiem in three sets, 6-4, 3-6, 6-3.
Murray found himself fending off a 20-year-old who is the youngest player ranked inside the top 125 and was a quarter-finalist last season in Kitzbühel and Vienna.
"It was an unbelievably tough battle," said the Scot. "There were a lot of long points. "He was hitting some huge shots, going for the lines. He had some really big winners. His game is fun to watch but it's no fun to play against at all. "I'm sure we will be seeing a lot more of him."
Murray started the evening with a double-break, but the contest turned into a struggle lasting almost two and a half hours.
Thiem recovered from the loss of the opening set to break in the eighth game of the second and level the match, before Murray gained some control back in the third set.
The Wimbledon champion won a monster second game in which Thiem saved three break points, and the contest finally ended seven games later on Murray's first match point.
The Scot now faces Marin Cilic, a former top 10 player whom he has defeated in 10 of 11 previous meetings.
Argentine Del Potro had it much easier on court but admitted that he is still competing with some pain in his fragile left wrist, which has been under the care of his doctor in Minnesota since late January.
"I know the wrist is getting better, I'm not worried about that. But I have to try and not think too much about it," he said. "Sometimes it gets frustrating, but the wrist problem is not dangerous so I can deal with it.
"The doctors are comfortable with how it's going - I can deal with a little bit of pain. It is to be expected after not touching a racquet for so long."
Elsewhere, Czech third seed Tomas Berdych posted another near-perfect showing, firing 10 aces and saving all four break points he faced against France's Nicholas Mahut on his way to a 6-4, 6-3 victory in just 72 minutes.
The Czech, who is playing the event for the eighth time, is looking to better his previous best showing, when he reached the semi-finals in 2012.
"I felt good on the court and I was hitting the ball very cleanly," said the world number seven. "I credit this win to my serve, which really helped me out. For me, serving was the key to the match."
Pole Jerzy Janowicz, a winner over German sixth seed Tommy Haas 6-4, 6-4, will take on Berdych next, while Del Potro will line up on Friday in the quarter-final against Latvian Ernests Gulbis, who upset eighth seed Grigor Dimitrov in straight sets.
German Philipp Kohlschreiber knocked out tired-looking fourth seed Richard Gasquet 7-5, 7-5 as the Frenchman was barely able to go through the motions.