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Argentina's victory moved the South American side top of the group, meaning they will face Group D runners-up the Socceroos in the next round, while second-placed Poland take on reigning champions France.
Argentina took the lead a minute into the second half, when Alexis Mac Allister got on the end of Nahuel Molina's cross and, despite making weak contact, saw his shot creep over the line with Polish keeper Wojciech Szczesny beaten.
The goal was just reward for Argentina who had dominated possession while Poland barely managed to get out of their own half.
The second goal was the result of some patient build-up, Argentina knocking the ball around before Enzo Fernandez unlocked the defence with a through-ball for Julian Alvarez, who found space in the box and smashed it into the top corner.
Earlier, Poland were up in arms when Argentina were awarded a controversial penalty after a VAR check for a foul on Messi - Szczesny's glove brushed the striker's face as he rose for a header at the far post.
But Szczesny was up to the task and despite the thousands of Argentina fans raising the decibel levels inside the arena, he kept his composure and guessed correctly, diving to his left and using one hand to swat aside Messi's effort from the spot.
Szczesny had been kept busy the entire half, with the Poland defence breached time and again but the Juventus keeper stood firm to keep his opposition at bay before the break.
But he could do nothing but scramble when Mac Allister took his shot before Poland had settled down after the second-half restart, while Alvarez's effort for the second goal was too good for any keeper.
