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Ralf Rangnick's Austria will likely secure qualification for the last 16 if they can avoid defeat in their final Group D game against the Netherlands on Tuesday.
Poland can no longer finish in the top two and need to defeat France to have any chance of progressing as one of the four best third-placed teams.
However, they will be out before that game if the Dutch fail to beat France later Friday.
"Since I've been Austria coach that's the most important match we have had," said Austria coach Rangnick.
"That's why we're so happy with the win... but also with the performance."
Gernot Trauner headed Austria into a deserved early lead before Krzysztof Piatek, who had missed their opening loss to the Netherlands, equalised for Poland in the 30th minute.
Lewandowski was introduced for his first appearance of the tournament with half an hour remaining, but was a peripheral figure as Poland slipped to the brink of elimination.
Christoph Baumgartner put Austria back in front after a fine team move and an end-to-end game was killed off by Arnautovic's 78th-minute penalty.
"The whole nation deserves this," said Baumgartner, who celebrated his goal with Rangnick.
"He (Rangnick) supported me every day, at half-time he told me that he needed me. So this is also a thank you to him."
Austria, who reached the knockout phase at a major tournament for the first time since 1954 at Euro 2020, will be sure of going through if they defeat the Dutch.
Lewandowski was not fit enough to start after suffering a thigh injury last week, despite Poland coach Michal Probierz saying he was hopeful the Barcelona striker would be able to play.
Fans from the two countries, which both border Germany, packed out the Olympiastadion.
The Poland supporters who had made the short trip outnumbered their counterparts, but it was the wall of fans in red Austrian shirts who were bouncing up and down in celebration less than nine minutes in.
Their team started in dominant fashion and were rewarded with the lead when defender Trauner powered a header past Polish goalkeeper Wojciech Szczesny after clever wing play from left-back Phillipp Mwene.
Poland grew into the contest, though, and Nicola Zalewski passed up a good opportunity to equalise when he blazed over.
The two most recent meetings between the teams, in Euro 2020 qualifying, produced only one goal.
Euro 2024: Poland 1-3 Austria
Piatek strikes
But an entertaining game this time was level just before the half-hour mark, as Piatek pounced to slot in from close range after Trauner blocked Jan Bednarek's shot.
Rangnick's side almost found themselves behind in first-half added time when goalkeeper Patrick Pentz was forced into a diving stop by Piotr Zielinski's free-kick.
Austria began the second half brightly, with Szczesny holding Stefan Posch's header.
A huge roar went up around the stadium as Lewandowski made his entrance in the 60th minute.
His only involvement of note was to pick up a yellow card shortly after coming on for catching Philipp Lienhart in the face with his elbow.
Austria moved back ahead with a wonderfully-worked goal in the 66th minute.
Arnautovic dummied substitute Alexander Prass' incisive pass and Baumgartner took a touch and coolly placed the ball into the bottom corner.
Austria could have wrapped up the three points with 15 minutes left when Patrick Wimmer's low strike was pushed to safety by the increasingly busy Szczesny.
But the third goal arrived shortly afterwards as Arnautovic calmly put away a penalty after Marcel Sabitzer bore down on goal and was felled by Szczesny.
Austria threatened to make the final scoreline embarrassing for Poland, with Szczesny making a brilliant fingertip stop from Posch's long-range drive.
The Poles were suddenly a shambles and Konrad Laimer dribbled a shot wide of an open goal after rounding Sczcesny, but even that miss could not take the gloss off Austria's third win at a major tournament in 34 years.
