Though the US Open champion three edged a close first set 7-5 on the tiebreak, he lost serve once in the second and three times in the third to make a premature exit from the ATP Masters 1000 event at the Indian Wells Tennis Garden.
Seventh-seeded Del Potro will next face world number one Novak Djokovic, who was barely tested as he demolished Jo-Wilfried Tsonga 6-3 6-1 earlier on Friday to extend his unbeaten run to 22 matches.
The first two quarter-finals were completed on Thursday, Rafa Nadal continuing his impressive comeback by thumping long-time rival Roger Federer 6-4 6-2 and Czech Tomas Berdych sweeping past big-serving South African Kevin Anderson 6-4 6-4.
"It was a tough match, I didn't serve particularly well," Murray said after serving eight aces and eight double faults during a match lasting just over two-and-a-half hours.
"There were a lot of long rallies. Very warm conditions the first set or two, and sometimes on the serve if your legs are just a little bit tired you can miss serves.
"Timing might go a little bit off and you're not quite getting up to them. That's maybe what happened today."
Asked if he had played the game he wanted to play against the 6ft-6in Del Potro, Murray replied: "I could've served better and returned better, two pretty important parts of the game."
Del Potro, the 2009 US Open champion, was delighted to beat Murray for the first time in five meetings on hard courts.
"I was close to beating him in 2009 in Montreal finals, but I did today," smiled the 24-year-old from Tandil after improving his record this year to 15-3. "I played my best match of the tournament.
"I was positive all the time, even when I lost the first set, a tough first set. In the end, I play my game. I be aggressive all the time... trying to make winners when I had the chance with my forehands."
On a sweltering afternoon at Indian Wells, Murray saved two break points in the first game of the match but there were no further opportunities for either player as a closely contested opening set went into a tiebreak.
A big forehand winner by the Scotsman earned him a 5-2 lead but he double-faulted and then netted a forehand, screaming to himself in disgust, as he let Del Potro pull back to 5-4.
The Argentine then hit a backhand long for Murray to earn his first set point at 6-4 and, though he squandered that with a wayward backhand, he clinched the set at the next opportunity on a backhand error by his opponent.
However, the Scot failed to take advantage and was broken to love in the second game of the second set after double-faulting.
Del Potro served out to level the match, clinching the set with a leaping overhead smash.
Murray was again broken in the third game of the final set, after netting a forehand, and also in the fifth, after hitting a backhand long, to trail 1-4.
Unable to break the towering Argentine's serve, Murray's challenge faded quickly and the match ended when the Scotsman double-faulted for an eighth time.