The hosts had never previously lost at home to their opponents, and set about extending that record through Robert Lewandowski, who had an early strike deflected narrowly wide.
Defender Elseid Hysaj then cleared Jakub Kiwior’s looping header off the line, but chances soon dried up, with a resilient defence holding firm and forcing the Eagles to shoot from distance, with Matty Cash and Filip Rozga unable to hit the target.
The Red and Blacks have never qualified for a WC, but they were growing in confidence and duly silenced the capacity crowd in Warsaw by taking the lead before the break. Jan Bednarek’s error - as he failed to cut out a through ball - allowed Arber Hoxha an opportunity to pounce, and he duly rounded the goalkeeper before calmly converting from a difficult angle.
That was a real wake-up call for the Poles, but they could not find a response before the interval, with Lewandowski whipping their best chance just wide of the far post.
With Jan Urban desperate for a revival, 17-year-old Oskar Pietuszewski was thrown on for his debut after the interval.
They were not far away from an equaliser when Zielinski curled off target from 22 yards, while Albania should have doubled their lead when Hysaj was at the forefront of a rapid counter-attack before feeding Nedim Bajramim, who blazed haplessly past the post from a central position inside the penalty area.
That could have put the match to bed, but instead the door remained open for a comeback, and Poland duly burst right through it when Thomas Strakosha flapped at a deep corner, allowing Lewandowski to climb highest at the back post and nod home from just inches out.
The goalkeeper did pull off an unconvincing stop from Zielinski minutes later, while opposite number Kamil Grabara denied debutant Stavros Pilios - who should have done better - at the other end.
That save proved crucial as the hosts hit the front in spectacular fashion with their next attack. Zieliński was left in acres of space 25 yards out, and he drilled a dipping strike into the far corner.
That shattered Albania’s resistance once and for all, as they lost on the road for the first time in a year, whereas Poland - who nearly grabbed a third when Pietuszewski fired wide - are through to face Sweden for a place at the WC on Tuesday.