Messi had a final chance to redeem his team with the final kick of the game, but his free kick from 25 yards failed to clear a solid defensive wall.
Goalkeeper Hannes Halldorsson was the unlikely hero after a phenomenal penalty save in a game that completely defied logic and expectations.
The lead lasted only four minutes with Alfred Finnbogason netting the equaliser, his country's first-ever World Cup goal.
True to form, Iceland fought back and Finnbogason netted the equaliser four minutes later, pouncing after Argentine keeper Willy Caballero parried Gylfi Sigurdsson's shot for a classic poacher's goal.
In the second half, Messi blew his chance at a Cristiano Ronaldo-style national saviour act when he once again made a hash of a penalty, awarded softly for Hordur Magnusson's innocuous nudge on Maxi Meza.
Sergio Aguero had given the twice world champions the lead after 19 minutes of the Group D match.
La Albiceleste boast one of the most potent selections of attackers at the World Cup, as demonstrated by their ability to start with Paulo Dybala and Gonzalo Higuain on the bench.
Messi stepped up but Iceland goalkeeper Hannes Halldorsson made the save to deny the Argentinian icon his first goal of the tournament. "It would have changed their game plan, too. This is going to be a tough group".More news: Comcast launches $65bn bid to steal Murdoch's Fox away from Disney
Iceland also protected Halldorsson well with swarming running and tackling to shut down Messi and Argentina's attacking lanes.
"That's just another statistic, it's part of the past", Argentina coach Jorge Sampaoli said of Messi's latest miscue. He said. "I tried to get in their minds, so they'd be thinking about me". They wrote one of the most lovely stories two years ago and they have full right to hope for the best this summer in Russian Federation with the same group of players on disposal.
Argentina, Iceland, Croatia and Nigeria compete in Group D. Argentina is favored to progress to the knockout stage, leaving Croatia, Nigeria and Iceland to battle for the final spot. "I looked at a lot of penalties from Messi and had a good feeling that he would go this way today".
Of course, downplaying Iceland so much would be a mistake after their antics at Euro 2016 and Argentina are now more than aware of that fact.
However, Iceland, the smallest nation ever to qualify, refused to roll over, with midfielders Aron Gunnarsson, Emil Hallfredsson and Birkir Bjarnason grafting like troopers to double up on Messi and shut down the spaces Argentina were trying to work.
However, Heimir Hallgrimsson's side was undeterred by the setback of conceding first.
Iceland, meanwhile, will believe that both Croatia and Nigeria are beatable as they look to build on their reputation as a burgeoning force in world football.
As we know, Portugal fought their way to glory two years ago and so Sampaoli's men can console themselves with the thought that teams can recover from an opening-game setback to go far.