St. Louis rallied from a three-goal deficit to tie the game when Glen Metropolit scored at 17:52 of the third.
Kovalev also had an assist; and David Aebischer stopped 25 of 28 shots for the Canadiens, who moved within one point of the last spot in the Eastern Conference playoff chase by breaking a four-game losing streak.
Doug Weight and David Backes also scored goals; Curtis Sanford allowed three goals on only 10 shots; and Jason Bacashihua gave the fourth goal for the Blues, who have lost three of four.
Metropolit had the game's only power play goal.
Celebrity Families of 2014 [PHOTOS]