Eller's goal at 6:09 of the third gave Montreal the lead for good and P.K. Subban added a power-play goal at 7:56 to make it 4-2.
The Canadiens, playing their third game of a five-game road trip, were outshot 44-28 but Carey Price made 42 saves to earn his 12th win of the season.
Carolina goalkeeper Justin Peters, playing his second straight game since being called up from Charlotte of the AHL, made 24 saves on 28 shots.
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