The Canadiens have won five out of six and eight out of 10, but remain a dozen points behind the leading Boston Bruins in the Northeast Division.
Guillaume Latendresse and Andrei Kostitsyn scored the other Montreal goals and Jaroslav Halak made 23 saves in his seventh consecutive start.
Kostitsyn put the Canadiens in front for good at 2-1 with a power-play goal midway through the second period.
J.P. Dumont and Vernon Fiddler scored for Nashville, which had won three out of four. Pekka Rinne made 17 saves in the loss.
Montreal played its final home contest before playing host to the NHL All-Star Game Jan. 24.