Kovalev's goal came just 43 seconds after the Blues tied it 3-3. It also snapped a four-game losing streak for the Canadiens, who have lost five of six.
Montreal jumped out to a 3-0 lead and chased Blues goalie Curtis Sanford early in the second period. But St. Louis came back and tied it with 2:08 left on Glen Metropolit's power-play goal off a rebound.
Doug Weight scored his 11th goal midway through the second to get St. Louis on the board and David Backes made it 3-2 with a goal seven minutes into the third period. The Blues thought they had tied it 3-3 a few minutes later when Lee Stempniak scored, only to have the referees wave it off for an offside call.