"We have to celebrate. It's a big thing for us but our focus is not just to take part," said Baumgartlinger after his team qualified for the first time since featuring as co-hosts with Switzerland at Euro 2008.
"We want to influence the championship in France next June...so we have to keep developing."
Tuesday's result meant Austria, with 22 points from eight games, made sure of finishing first in Group G with two qualifying matches to spare.
The visitors swarmed forward whenever they had the chance, overwhelming the Swedes with their quick passing to hand coach Erik Hamren's men their heaviest defeat at the stadium.
"We're very relieved, we thought we had a good position in the group," said Baumgartlinger who plays his club football in Germany's top flight with Mainz 05.
"We were really trying to focus, to win it today and not in the games in October, and that's what we did."
Martin Harnik netted a goal in each half, David Alaba chipped in a first-half penalty and Marc Janko also got on the scoresheet before Zlatan Ibrahimovic grabbed a late consolation for Sweden.
Beaming with delight after the victory and the long, loud celebrations with the travelling Austrian fans, Baumgartlinger said his team had quickly abandoned any idea of playing it safe in Stockholm.
"We were determined to win here, we really wanted it from the beginning," the 27-year-old said.
"We tried something different, playing a little bit deeper and more defensively, but after five or 10 minutes the Swedes were getting a lot of crosses in so we felt we had to move higher up and be more aggressive.
"The penalty went our way and after the second goal in the first half we felt that we were not going to let this be taken away from us."
Sweden's defeat was their first at home in 14 European Championship qualifiers.
"We have great quality in the team. Every player in the 23-man roster is developing every week, every year," said Baumgartlinger as he looked ahead to the Euro 2016 finals in France.
"We have a lot of players playing in good leagues. Our tactics are modern and not easy to play against. We're difficult opponents."
(The story has been refiled to make clear Austria previously qualified as co-hosts)
(Editing by Tony Jimenez)
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