Berggren opened the scoring on the power play at 7:12 in the first period, giving the Red Wings an early 1-0 advantage.
"It was fun to score my first NHL goal," said Berggren, who was recalled from the American Hockey League's Grand Rapids Griffins last Thursday. "It was an unreal feeling when I saw the puck go in."
The Ducks responded with a power-play goal later in the opening frame, then Michael Rasmussen's second-period tally put Detroit ahead, 2-1, after 40 minutes. Anaheim tied it, 2-2, with 47 seconds left in regulation, then recorded the game-winner in the final minute of overtime.
"It stings right now," captain Dylan Larkin said. "(John) Gibson played really well. (Ville) Husso played well. Just a 2-2 game, we gotta find a way on one of those power plays to go out and finish the game."
Ville Husso made 26 saves for the Red Wings (7-5-4; 18 points), who have lost four straight contests.
Jake Walman made his season debut on Tuesday, recording a plus-1 rating in 13:29 of ice time. Tyler Bertuzzi, who had missed the previous 13 games due to an upper-body injury, logged three shots in 18:39.
NEXT UP:Detroit will play the third game of its four-game road trip on Thursday night, facing the San Jose Sharks at SAP Center.