Jonatan Berggren
scored his first NHL goal, Michael Rasmussen had a goal, and Ville Husso made 27 saves for the Red Wings (7-5-4), who have lost four in a row (0-2-2).
"We had a good flow to our game through two periods and, unfortunately, we didn't manage our third very well," Detroit coach Derek Lalonde said.
The Ducks forced a turnover by Tyler Bertuzzi in the Red Wings zone and Trevor Zegras passed to Strome for the tap-in. Bertuzzi missed the previous 13 games because of a broken hand.
"We had a really good start to our season and we've been competitive in this league because we take risk out of our game but, unfortunately, a couple times in the third and especially in overtime we got away from that," Lalonde said.
Klingberg scored on a point shot through traffic to tie it 2-2 with 47 seconds left in regulation and Gibson pulled for the extra attacker. It was Klingberg's first goal with the Ducks and the second of the season by an Anaheim defenseman.
"That's how you score goals nowadays; it's tough to make perfect plays," Klingberg said. "That puck found its way all the way down there and it went in, so it's always fun to score goals."