Goalkeeper Szczesny's double save from a penalty in first-half stoppage time kept Poland in the match against a Saudi side that was always on the attack.
Poland has bolstered their chances of reaching the World Cup's last 16 for the first time since 1986 after goals from Piotr Zielinski and Robert Lewandowski blocked Saudi Arabia's efforts to advance a game early.
Zielinski scored after 39 minutes on Saturday, against the run of play and Lewandowski put the match beyond Saudi Arabia eight minutes from time.
Lewandowski, who missed a penalty in Poland's opening draw with Mexico, was visibly emotional after finally finding the net, scoring his first ever World Cup goal.
Goalkeeper Wojciech Szczesny's double save from a penalty in first-half stoppage time kept Poland in the match against a Saudi side that was always on the attack.
The result leaves Poland needing only a point from their final Group C match against Argentina to guarantee their spot in the next round.
Saudi Arabia were the better team for the first half but Salem al Dawsari, who scored the winner in their historic victory over Argentina on Tuesday, missed a penalty against Poland.