The match at Stadium 974 in Doha saw Poland fight hard to avoid an early exit from the global tournament for the second straight time after a disappointing 2018 World Cup in Russia.
Alexis Mac Allister put Argentina 1-0 up in the 46th minute of the game and Julian Alvarez doubled the score in the 67th.
Poland goalkeeper Wojciech Szczęsny saved a penalty from Argentina great Lionel Messi in the 39th minute.
Argentina played their best football of the tournament and controlled the game from start to finish.
The Polish team finished second in their Group C and will face world champions France in the round of 16 on Sunday.
Poland progressed to the knockout stages of the World Cup for the first time in 36 years, with four points from three group-stage games.
Argentina finished at the top of the group, with six points.
Meanwhile, Mexico and Saudi Arabia crashed out of the tournament after finishing third and fourth respectively.
Poland, captained by star striker Robert Lewandowski and coached by manager Czesław Michniewicz, tied 0-0 with Mexico in their opening game at the World Cup in Qatar on November 22 and then beat Saudi Arabia 2-0 in their second fixture four days later.
Poland are ranked No. 26 worldwide by world football's governing body FIFA. Meanwhile, Argentina are in third place; Mexico are ranked 13th, and Saudi Arabia are at No. 51 in the FIFA table.
Teams from 32 nations will play a total of 64 matches at eight stadiums in Qatar during this year's World Cup, which kicked off on November 20 and runs until December 18.
According to a study by experts at London's Alan Turing Institute, Brazil are in the best position to win the World Cup in Qatar, while Poland are the 16th-most likely winner.
Poland qualified for the 2022 World Cup after defeating Sweden 2-0 in a playoff at home in late March.
The Polish team failed to advance to the knockout phase of the 2018 World Cup in Russia after losing 1-2 to Senegal and 0-3 to Colombia in the group stage. Poland won their final game, beating Japan 1-0, but finished bottom of their Group H and made an early exit from the tournament more than four years ago.
Poland finished third in the 1974 World Cup in West Germany and were also third in Spain in 1982.
(gs)
Source: TVP, IAR, PAP