Evgeny Svechnikov, defenseman Filip Hronek and Vladislav Namestnikov also scored for Detroit, which beat Columbus for the second time in two days. The Red Wings won 3-1 on Saturday.
''This was probably our most complete effort from start to finish this season,'' Detroit coach Jeff Blashill said. ''Certainly, I thought Saturday was pretty good too, and the first period was great, but I just thought we sustained our forecheck more today and because of that, the ice was tilted toward them for large parts of the game.''
Calvin Pickard made 16 saves and got both wins.
''We competed all over the ice, and when they did get a couple chances, Pixie (Pickard) was back there,'' Red Wings captain Dylan Larkin said. ''He's just been rock solid this weekend. He didn't give up rebounds and just swallowed everything up. He's really been the X-factor the last two days.''
Jack Roslovic scored for the Blue Jackets. Elvis Merzlikins stopped 25 shots.
''We played awful. ... We didn't play well,'' Columbus coach John Tortorella said. ''I think we played harder today (than yesterday), but couldn't make a play.''
Rasmussen's winner came 2:51 into the third period when he put in the rebound of Robby Fabbri's shot. The Blue Jackets challenged that there was goaltender interference on the play.
Hronek and Namestnikov scored into an empty net in the final two minutes. It was Hronek's second goal and the sixth for Namestnikov.
Roslovic tied the game 1-1 with 1:35 remaining in the second period when he tipped Oliver Bjorkstrand's shot out of mid-air and past Pickard. It was Roslovic's sixth goal and snapped Pickard's shutout streak of 66:42.
Svechnikov gave Detroit a 1-0 lead with 9:20 left in the first period when he beat Merzlikins with a wrist shot from the high slot as Svechnikov skated left to right. It was Svechnikov's third goal.
NOTES: Detroit forward Bobby Ryan left the game after the first period with an upper-body injury. There was no update on him after the game ... Roslovic's goal was his first point in four games.
Blue Jackets: at Tampa Bay Lightning on Tuesday.
Red Wings: at Florida Panthers on Tuesday.
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