The Paris St Germain striker latched on to Kim Kallstrom's long ball and slotted home with four minutes to go to make sure Sweden finished as runners-up to group winners Germany.
"It's magical," Ibrahimovic told Swedish TV after the game. "It's a fantastic feeling. We are a step closer to the World Cup."
Germany head the group with 25 points going into the final round of qualifiers on Tuesday. Sweden have 20 points and will go into the playoffs for a place in Brazil next year. Austria are third with 14 points.
Ibrahimovic's side had to dig deep to come from behind after the visitors took a 29th-minute lead.
Marko Arnautovic surged clear down the left and his cross was touched on by Marc Janko for Martin Harnik to thump the ball home.
Johan Elmander hit the post for Sweden before half-time but it was a different story after the break as Ibrahimovic began to exert a strong influence.
The home side levelled in the 56th minute, left back Martin Olsson meeting Ibrahimovic's deft chip with a diving header.
Sweden continued to threaten, Olsson firing a left-footed drive just over as they pressed for a winner.
The game looked to be heading for a draw until substitute Kallstrom set up Ibrahimovic.
"The whole team should have credit, not only me, because we came out totally different in the second half," Ibrahimovic told reporters.
He will miss the final game of the qualifying campaign, at home to Germany on Tuesday, after picking up a late yellow card.
(Editing by Tony Jimenez and Stephen Wood)
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