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Del Potro won 6-4 5-7 6-7 (5-7) 6-3 6-4 after five hours and seven minutes, before Guido Pella saw off Kyle Edmund 6-7 (5-7) 6-4 6-3 6-2.
Murray's defeat was only his third in 32 Davis Cup singles matches.
GB must win Saturday's doubles and both Sunday's singles to reach the final.
Murray is scheduled to team up with his brother Jamie for the doubles contest but admitted it would be "really tough" for him to recover following the longest match of his career on Friday.
The winners of the best-of-five tie will face France or Croatia in the final, with that semi-final level at 1-1 after day one.
The world number two hit 35 aces and battled through the longest match of either man's career, but Murray came up short against an inspired Del Potro, who lost their four-hour Olympic final last month.
"I won but I still remember the Olympic final," said Del Potro. "This is very important to me and my team and we are looking to reach another final in the Davis Cup.
"Against Andy you never know when the match will finish. He is a dangerous player, a fighter and great champion. In end I played good forehands and good serves and that was key."
Murray said: "I think it was good match. I felt there wasn't too many mistakes on important moments.
"Both of us came up with good stuff and the shot he came up with to break in the fifth set was the difference."
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Del Potro hit 26 forehand winners and broke serve six times to join Stan Wawrinka and Fabio Fognini as the only men to have beaten Murray in Davis Cup singles matches.
The Argentine, ranked 64th as he works his way back after three wrist surgeries, was the better player for the first two hours and earned a break point that would have left him serving for a two-set lead.
Murray clung on, however, levelling at one set all after breaking serve for 7-5, following a call from the crowd on set point that incensed Del Potro and Argentine captain Daniel Orsanic.
Del Potro's mix of slow, sliced backhands and huge forehands continued to trouble Murray, but the Wimbledon champion twice came back from a break down in the third before saving set point with a stunning forehand lob.
It was a shot that electrified the Scot and his home crowd, and he would edge ahead thanks to an attacking return at 5-5 in the tie-break followed by a big serve.
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Both men were unhappy with a succession of line calls and umpiring decisions, with Murray at one stage saying: "This is getting ridiculous now, it's actually getting ridiculous. It's so, so bad."
A loose Murray service game at 2-1 down was all that separated them in the fourth set, and with fatigue taking a grip they began a decider with the match entering its fifth hour.
Both players had their trainers on at the changeover to try and work some energy into their legs, and it was Del Potro who made the first push at 2-2 only for Murray to save break point with a big serve and a shout of "Let's go!"
Del Potro got the decisive break two games later with a rasping forehand winner after Murray had put a forehand volley within the Argentine's reach, and served out to the delight of the 400 travelling fans.
"I am so happy to be playing tennis again after three surgeries," said Del Potro.
"This kind of moment was what I was missing at home and I always play good in Great Britain, and here I played great, so it is a good place for my tennis."
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After the drama of the opening match it looked as though the similarly fearsome forehand of Edmund might draw the hosts level in a more low-key second contest.
The 21-year-old came back from 3-0 down to win the opening tie-break and had a break point early in the second, but Pella worked his way into the contest.
Five years older and ranked six places higher at 49th in the world, Pella's steadier play prevailed as Edmund went on to make 36 winners but 50 errors.
Pella took the second set with the lone break in game 10 and moved ahead with four straight games in the third.
Captain Smith could be heard urging on his number two player but Pella remained the dominant force in the fourth set, breaking serve at the start and going close to extending his advantage as he won in just over three hours.
"I am happy to give the team a second point after Juan Martin's amazing match," said Pella. "I was fighting every point no matter what."
Captain Leon Smith said it was one of his team's "rougher days" in the competition after enjoying "a lot of good days" in the past but he stressed the team "have to keep fighting".
He added: "Credit to Andy and Juan for a match of high quality between two of the best players in the world right now.
"It's the same for everyone, particularly Andy and Juan who win lots of matches and go deep into tournaments, but playing the Davis Cup at this time of year, a stage of the season where a lot of tennis has been played already, is difficult."
Russell Fuller, BBC tennis correspondent:
Before the opening day of this semi-final, Andy Murray had not lost a Davis Cup match since April 2014. The symbolism of Juan Martin del Potro's opening victory was not lost on the rest of the Argentine team, and Britain will now have to win from 2-0 down - as they did against Russia in the 2013 tie which sparked their return to the World Group.
Murray's five hour, seven minute match against del Potro is believed to be the longest ever completed by a British singles player. He and Leon Smith have until an hour before the start of Saturday's doubles to decide whether Dan Evans might, on this occasion, be a better partner for Jamie Murray.
Great Britain's Davis Cup defence is hanging by a thread after Juan Martin del Potro beat Andy Murray to set Argentina on their way to a 2-0 lead on day one of the semi-final in Glasgow.