Poland earned a crucial victory over Saudi Arabia that leaves the group wide open.
On an error by Saudi Arabia's Abdelulelah Al Malki, the Barcelona forward capitalised on his chance and sent a finish past Mohammed Al Owais.
After missing a penalty in Tuesday's goalless draw with Mexico, Poland's Robert Lewandowski broke down in tears.
Saudi Arabia were awarded a penalty five minutes after Poland had taken a first-half lead.
Szczesny stopped Mohammed Al Burayk from burying the rebound after Feras Al Brikan tried to score.
In the second half, Saud Abdulhamid, Al Brikan and Arkadiusz Milik were all denied by the woodwork, as Poland sought to double their lead on the counter- attack.
Poland's hopes of reaching the last 16 for the first time in 26 years have been boosted by the late strike of Robert Lewandowski, which ended Saudi Arabia's hopes of building on Tuesday's victory over Argentina.
Saudi Arabia and Mexico will play for a place in the last 16 of the World Cup.
Saudi Arabia will be angry not to have preserved their perfect start to the World Cup after finding Sczcesny in inspired form.
Backed by a fervent and partisan crowd inside Education City Stadium, Saudi Arabia applied heavy early pressure and were denied a great opener when the formerArsenal goalkeeper palmed Mohammed Kanno's ferocious drive over the bar.
Poland looked increasingly rattled as the Green Falcons continued to swarm forward in waves, but Bielik's goal bound headed was cleared to safety by Al Shehri before Zielinski gave them the lead.
After reviewing the incident on his pitchside monitor, the referee from Brazil pointed to the spot after initially ignoring the challenge.
If Szczesny's save from Al Brikan's spot-kick was good, his save from Al Burayk's follow-up effort was even better.
The pattern of play in the second half was the same as in the first, with Szczesny saving at the feet of Abdulhamid and Al Brikan.
Poland continued to be a threat on the counter-attack as Milik headed against the bar and Lewandowski's shot came back off the woodwork.
Saudi Arabia's hopes of making it to the last 16 were ended by Al Malki's heavy touch.
In the final moments of the game, Al Owais was able to keep out the goal attempt from the strikers.
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