Second-half goals from Alexis Mac Allister and Julian Alvarez were plenty for Argentina, who were shocked by Saudi Arabia in their group opener, to keep their quest for a first World Cup title since 1986 alive.
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"It's so emotional for me, for the whole squad. It's a dream come true," Mac Allister told reporters. "I debuted with this team, I'm so proud of that. And now managing to score in a World Cup. I'm so happy.
"[After Messi's penalty miss] we tried to stay positive, to stay calm. We did not feel down. We tried to be optimistic. Fortunately we managed to win and go through."
Despite being thoroughly outplayed through the 90 minutes, Poland also snuck into the knockout round in second place thanks to a superior goal difference over Mexico and will face Group D winners France on Sunday. Argentina is set to face Australia on Saturday with a spot in the quarterfinals on the line.
Messi was playing in his 22nd game at the World Cup -- and 999th of his career for club and country -- breaking a tie with Diego Maradona for the most by an Argentine in the world's biggest tournament.
Poland goalkeeper Wojciech Szczesny saved a Messi penalty that was awarded by VAR as Argentina dominated the first 45 minutes but had nothing to show for it with the halftime score 0-0.
Szczesny had been kept busy the entire half with the Poland defence breached time and again, but the Juventus keeper stood firm to deny the Argentines who grew more confident with every attack.
He first denied Alvarez, who broke through the offside trap before he tipped Angel Di Maria's cross over the bar when his Juve teammate attempted to score directly from a corner kick.
The opener for Argentina came right after the restart when Brighton striker Mac Allister redirected a Nahuel Molina cross with a first-time shot that trickled past Szczesny and off the far post.
Manchester City teenager Alvarez announced himself on the world stage and doubled Argentina's lead midway through the second half, picking up a pass from Enzo Fernandez in the penalty area and lashing a shot past Szczesny to all but secure the three points.
With Poland's spot in the round of 16 looking like it could come down to a fair play (yellow and red card accumulation) tiebreaker with Mexico, Poland coach Czeslaw Michniewicz had instructed his players to avoid unnecessary bookings as the match wore on.
"I didn't tell players to take their foot off the ball, but I did tell them to avoid silly bookings," Michniewicz told a news conference. "We even talked about the yellow cards for those on the bench, because it was only before the match, after the meeting with the delegate, when we learned that cards awarded to players on the bench do not count for the fair play classification.
"We were afraid that one of the players would jump out, or the coach, and would get booked and therefore we might not advance further. But I also asked that we do not provoke, not pull their shirts, not argue with the referee."