Alexis Mac Allister’s 46th-minute goal settled Argentina‘s nerves after a tense first half, before Julian Alvarez doubled their lead just over 20 minutes later.
Argentina had been unable to find a way past Wojciech Szczesny in an opening 45 minutes in which the Poland goalkeeper saved his second penalty of the tournament.
Szczesny was adjudged to have fouled Messi while stretching to reach a cross, but he dived to his left to beat away the resulting spot-kick.
It was the 31st penalty Messi had failed to convert in his career.
Poland offered little attacking threat, with star striker Robert Lewandowski virtually anonymous throughout.
Instead it was left to Szczesny to take centre stage for the Poles, denying Messi with two crucial early stops.
First Messi wriggled clear of a pack of defenders before producing a shot that flew straight at the keeper.
Then he found space to charge into the Polish box down the left, forcing Szczesny to make a save at his near post.
Argentina continued to create the better chances in the first half and Marcos Acuna flashed a shot wide after Alvarez had also gone close to breaking the deadlock.
Szczesny had to scramble to stop Angel di Maria scoring direct from a corner, before the goalkeeper was again called into action to deny Alvarez.
Szczesny outdid himself when he dived to his left to stop Messi’s penalty in the 39th minute.
He still had more to do before the half-time whistle, punching away a shot from Alvarez before saving the follow-up from Rodrigo de Paul.
There was nothing he could do when Mac Allister broke the deadlock less than a minute after the interval.
Nahuel Molina charged down the right, then cut the ball back for Mac Allister to drill just beyond Szczesny’s reach.
The goal forced Poland to show some belated attacking ambition and Kamil Glik headed just wide from a free-kick almost immediately after the restart.
Argentina continued to go in search of more goals and Alvarez scored their second after combining well with Enzo Fernandez.
Fernandez, making his first start for Argentina, picked out his team-mate, who held off a defender before lashing the ball past Szczesny.
Poland were still walking a tightrope with yellow cards potentially crucial to their chances of reaching the last 16.
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