The closest Messi came was a half-chance, a snatch at a rebound from a Maximiliano Meza shot which was blocked by Danijel Subasic.
Lionel Messi and Argentina are facing the very real prospect of an early World Cup exit after a dreadful error from goalkeeper Willy Caballero set Croatia on their way to victory and a place in the last 16.
On a night when so much was expected of Messi, Chelsea reserve goalkeeper Caballero, who made the most telling contribution to his team’s fate with an attempted chip over Ante Rebic that backfired badly, allowing the striker to fire home into an exposed net in the 53rd minute.
Messi – adrift for much of the match, especially during a pedestrian opening half – helped to rally his side after going behind, but neither he nor substitute forwards Gonzalo Higuain and Pablo Dybala could find a way through.
The closest Messi came was a half-chance, a snatch at a rebound from a Maximiliano Meza shot which was blocked by Danijel Subasic and turned behind to safety.
It was a dogged defensive second-half display from Croatia, illuminated by a moment of magic from their own masterful number 10, Luka Modric.
His curled shot from range drifted beautifully beyond Caballero’s dive to seal a victory that took Croatia through to the knockout stage and left them in a strong position to win the group.
Ivan Rakitic rounded off a miserable evening for Argentina as he rolled in Croatia’s third goal in injury time.
For Argentina, they are no longer in control of their own destiny.
This defeat follows their opening 1-1 draw with Iceland, in which Messi missed a penalty, and now even a heavy win over Nigeria in their final game on Tuesday may not be enough to send them through to the next round.
Should Iceland beat Nigeria in their second group game on Friday, Iceland would need only a draw against Croatia in their final match.
For Barcelona forward Messi, Argentina captain and a prolific performer at club level, this was just another harrowing chapter in his recent history of pain and disappointment in the international game.