Mikael Samuelsson scored the winner early in the third period to snap a 1-1 tie as Sweden improved its record to 3-1.
The United States drops to 1-2-1 with their only win so far coming over lightly-regarded Kazakhstan.
Forsberg dressed for the game but ended up sitting on the end of the bench with his helmet off. He is suffering from a groin injury and it was only on Friday that he decided to suit up for the squad.
On Samuelsson’s goal, Mats Sundin started the play by winning the draw. Sundin got the puck back and sent it to the front of the net where Samuelsson was waiting.
Samuelsson kicked the puck from his skate to his stick in one motion and slapped it past American goalie Rick DiPietro at 4:22 in the final period.
Daniel Alfredsson scored the first Swedish goal at 7:05 of the opening period to make it 1-0.
Mike Modano tied it just over 10 minutes later for the Americans who have not beaten Sweden in their last eight meetings dating back to 1964.