Gauthier gave Anaheim a 4-3 lead at 11:09 of the third on a wrist shot off the rush.
"It was a dream come true," said Gauthier, who was playing in his 17th career game. "Against the hometown team, Detroit, it was a surreal moment. I'm just so pumped."
Gauthier started the season on the top line but had dropped to the fourth line after he struggled in the early going.
"It's the best league in the world and you don't have a lot of time and space," Gauthier said. "I think the biggest thing I've noticed is when to manage the puck, when to make the right play when I have some time, and when I've just got to keep it simple and get it deep and get on the forecheck."
Olen Zellweger had a goal and two assists, and Trevor Zegras and Troy Terry each scored on the power play for the Ducks (6-8-2), who had lost five of six. John Gibson made 21 saves.
Marco Kasper scored his first NHL goal and had an assist, and Alex DeBrincat and Lucas Raymond each also had a goal and an assist for the Red Wings (7-8-1), who have lost three of four (1-3-0). Alex Lyon made 23 saves.
"We got away from what got us to the 3-1 lead, and then we took penalties," Detroit coach Derek Lalonde said. "It led to three straight penalties, and then it felt like (the Ducks) started to play downhill. So, unfortunately, I think it turned once we had the 3-1 lead."