Poland's Piotr Zieliński scored the first goal in the 39th minute, and the team's star striker Robert Lewandowski found the back of the net in the 82nd to seal the result.
Meanwhile, Poland's goalkeeper Wojciech Szczęsny saved a penalty in the 45th minute of the match, helping Poland claim the win.
Lewandowski, Poland's all-time top scorer, bagged nine goals in his team’s World Cup qualifying campaign, but the goal against Saudi Arabia was his first career World Cup goal after failing to score at the 2018 tournament in Russia.
The 34-year-old Barcelona striker has scored 77 goals in 136 appearances for Poland.
After their victory at the Education City Stadium in Al Rayyan near Doha, Team Poland took the lead in their group C.
Poland tied 0-0 with Mexico in their first game at the World Cup in Qatar on Tuesday.
Poland boss Czesław Michniewicz said after the game the team was "happy to bring so much joy to all our fans here in Qatar, and back in the country."
"I think we are all enjoying these beautiful moments," Michniewicz added.
Michniewicz's men are scheduled to face off against Argentina at Stadium 974 in Doha in their final group-stage fixture on November 30.
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 Sunday 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.
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Source:PAP