Lewandowski's punishment from the get-go in the last part, after a foul on Grzegorz Krychowiak, broke the halt.
Zielinski exploited a cautious pass to run clear and settle the tie for the home side.
The draw for the finals happens this Friday in Doha with the World Cup getting going on 21 November.
It will be whenever the competition first has been moved from its standard summer opening, for games to be played in milder winter temperatures in the Gulf.
Furthermore, Poland will be there after a hard-battled triumph over a Sweden side that had a tremendous impact in denying them a spot in the knockout phases of Euro 2020.
Best activity from Tuesday's World Cup play-off finals, including Poland's success over Sweden
It was accomplished with various players conveying wounds as well, as per Lewandowski.
"We had numerous issues before this match," he said. "A great deal of our players were harmed, including me - I disapproved of my knee.
"We endured, yet presently we can celebrate. I'm pleased and blissful."
Protector Kamil Glik, who played with a muscle injury, said: "It was new for us all - as far as I might be concerned, for Robert [Lewandowski], who had played many matches however he hadn't played one like this, not a single one of us had.
"I wasn't keen on what might happen tomorrow, the day after tomorrow, one week from now, one year from now. All that made a difference was the present time and place."
It was practically inescapable that Bayern Munich striker and Poland charm Lewandowski would play a significant part in the game - thus it demonstrated.
He was tranquility represented in hiding his spot-kick after Jesper Karlstrom's ungainly test on substitute Krychowiak in the container.
It was a sledge disaster for Sweden who had been the better side and scorned the most obvious opportunity with regards to the principal half when Emil Forsberg shot excessively near guardian Wojciech Szczesny who saved.
The guests revitalized subsequent to falling behind, with Forsberg again testing Szczesny, yet saw the tie get away from them when Marcus Danielson's unfortunate control at the back permitted Zielinski to nip in, run in on objective and twofold the lead.
Zlatan Ibrahimovic was acquainted with 10 minutes with go, proposing the forward was not completely fit, yet it was short of what was needed.
The 40-year-old AC Milan forward has now probably played his last World Cup finals match.
In the wake of experiencing a late capitulation to pass up programmed capability through the gathering stage, 2018 World Cup quarter-finalists Sweden have now experienced a definitive knockout blow.
Yet, Poland will get the opportunity to enhance their gathering stage disposal in Russia four years prior.
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