Joe Veleno and Dylan Larkin also scored and Alex Lyon made 29 saves as the Red Wings completed a successful three-game California road trip to San Jose, Los Angeles and Anaheim.
The three victories to start January match what Detroit had in its final 13 games of December.
Rasmussen’s game-winner, his ninth goal of the season, came when he redirected the puck off his left skate and into the Ducks’ goal off a pass from Moritz Seider. The play was reviewed with the goal allowed after no kicking motion was detected.
Trevor Zegras scored two goals for the Ducks and Lukas Dostal made 25 saves as primary goalie John Gibson was out with an illness. Troy Terry had an assist for Anaheim after missing the previous three games with an upper-body injury.
The Ducks went 1-6-1 on a season-long eight-game homestand and now head out for six consecutive road games.
The Ducks looked content to finish the homestand on a high note when they scored 20 seconds into the game. Terry fired a shot into traffic and Zegras deflected the puck toward the ice and into the goal after it took an indirect bounce past Lyon.
The Red Wings tied it at 11:10 of the first period while on the power play when Larkin scored from the slot off a pass from David Perron for his 14th of the season.
Detroit grabbed a 2-1 lead at 2:20 of the second period when Veleno scored on a wrap-around that he put up against Dostal’s pad and just across the line at the right post. Veleno’s eighth of the season was confirmed on replay.
Zegras tied it at 3:55 of the third period when he took a cross-ice pass from Cam Fowler near the edge of the right circle and ripped a shot into the top right corner of the goal for his fourth of the season. With the assist, Fowler tied Steve Rucchin for fifth most points in Ducks history (432).
The Ducks survived a Red Wings power play with just under four minutes remaining before giving up the go-ahead goal with the teams at full strength.
Anaheim was without forward Max Jones, who was placed on injured reserve with an upper-body injury.
Up next for the Ducks: At Nashville on Tuesday