Substitute Quaison opened home after a one-two with Alexander Isak.
Poland, who got a bye after semi-last adversaries Russia were restricted from World Cup qualifying following the attack of Ukraine, have Sweden on Tuesday with the victor going to Qatar.
Manchester United youngster Anthony Elanga came on for his Sweden debut late on at Friends Arena in Solna.
It was anything but a game with too some obvious possibilities, however Isak and Tottenham's Dujan Kulusevski had endeavors for Sweden in the principal half.
In the last option phases of the game, Sweden's Mattias Svanberg going directly at the Czech goalkeeper from Emil Forsberg's corner.
Tomas Pekhart might have won it for the Czech Republic in injury time toward the finish of the hour and a half when he was plain in the crate, yet all at once volleyed wide.
Sweden striker Zlatan Ibrahimovic will be accessible for the last against Poland, having been suspended for this game. He last played in a World Cup in 2006.
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