
Halftime substitute Sabitzer converted his spot kick after 48 minutes following a handball by defender Rahil Mammadov in a game largely dominated by the visitors.
The victory ensures a top-two finish for Austria along with already-qualified Belgium, the third time in a row the Austrians will be present at the continental finals, and fourth overall.
The result will likely spark the resignation of Sweden coach Janne Andersson, who said in a federation statement earlier this month that he would step down after seven years in charge if his side failed to qualify for the tournament.
Sweden play away to Belgium later on Monday.
Austria are the eighth side to book a place at the June 14-July 14 finals along with Germany, as hosts, Belgium, France, Portugal, Scotland, Spain and Turkey.