The two-time world champions have been pushed out to 33/1 by Sky Bet to lift the World Cup having collected just one point from their first two games against Iceland and the Croatians, who are now 14/1 with the online bookmaker having advanced to the knock-out phase thanks to second-half goals from Ante Rebic, Luka Modric and Ivan Rakitic.
Rebic volleyed in just after the interval when Chelsea's reserve goalkeeper Willy Caballero inexplicably chipped an attempted pass straight to him, with Modric scoring a second with a brilliant strike from range 10 minutes from time and Rakitic placing in a third late on.
Messi, who had a penalty saved in Saturday's 1-1 draw with Iceland, was again below-par and saw Rakitic block his only real opportunity across the 90 minutes.
Argentina boss Jorge Sampaoli had made three changes from the draw with Iceland but his team were all at sea throughout in their 3-5-2 formation, with the South Americans fortunate Caballero denied Ivan Perisic when he was allowed to glide into the box early on.
Croatia looked vulnerable at the back too, yet Enzo Perez placed his attempt wide when left with a virtual open goal to aim at on the half-hour mark.
Mario Mandzukic was also guilty of spurning a glorious opening as he reached a back-post cross undetected only to head wide from close range.
The contest was becomingly increasingly tetchy and Rebic was perhaps fortunate his studs-up challenge on Eduardo Salvio led to a yellow card, rather than a red shortly before half-time.
He was allowed to stay on the pitch, though, and was in the right place when Caballero made his gaffe eight minutes after the break.
The stopper had been attempting to dink a pass to defender Gabriel Mercado, but the ball looped straight up into the air and was emphatically volleyed back over Caballero's head by Rebic.
Juventus pair Gonzalo Higuain and Paulo Dybala were then brought on as substitutes, and Argentina nearly forced an equaliser only for Croatia goalkeeper Danijel Subasic to deny Maximiliano Meza and Rakitic to thwart Messi with a brilliant late tackle from the rebound.
Modric then made sure of Croatia's victory with 10 minutes to go as he worked space on the edge of the box and bent a brilliant attempt inside Caballero's post.
Argentina's frustration was evident as Nicolas Otamendi kicked the ball into Rakitic after he won a free-kick, which the Barcelona midfielder then dispatched against the frame of the goal.
Rakitic would get his goal in stoppage time, though, collecting a pass from substitute Mateo Kovacic to roll in with Argentina's midfield and defence nowhere to be seen.
Opta match statistics
- This was Croatia’s first victory in five attempts against South American nations at the World Cup, having lost each of their previous four.
- This was Argentina’s heaviest defeat in the first round group stages of a World Cup tournament since losing 1-6 to Czechoslovakia in 1958.
- This was Croatia’s joint second biggest World Cup victory. They beat Cameroon 4-0 in 2014 and Germany 3-0 in 1998.
- Argentina are now winless in their last four World Cup games (D2 L2), their longest ever such run in the competition.
- Argentina’s Lionel Messi – who had 11 shots in the opening game against Iceland – didn’t manage a single attempt in goal until the 64th minute of this match.
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Odds correct as of 2050 BST 20/06/18
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