Kovalev beat former teammate Carey Price with a shot from the slot 11:28 into the third period after he set up Ottawa captain Daniel Alfredsson's power-play goal during a two-man advantage in the second.
Chris Neil also scored and Pascal Leclaire made 27 saves as the Senators (5-2-0) won for the fifth time in six games.
Mike Cammalleri scored his first goal in seven games with Montreal, which has lost five straight since winning its first two games of the season in overtime.
Canadiens goalie Carey Price stopped 18 shots.
Alfredsson got his third goal of the season during a 5-on-3 advantage 4:44 into the second to put Ottawa up 2-1. He fired a shot past Price into the left side of the net off a feed from Kovalev, who patiently held onto the puck on the right side, the kind of play the crowd of 21,273 had seen him make numerous times during his stint with the Canadiens.
Kovalev, who was the MVP of the 2009 All-Star Game at the Bell Centre, left Montreal to sign a two-year, $10 million contract with Ottawa on July 6.
Prior to that, dozens of fans organized outside Bell Centre this summer, pleading with Canadiens general manager Bob Gainey to re-sign Kovalev.
A loud cheer went up when his name was called by PA announcer Michel Lacroix following his third goal of the season, the 397th of his 17-season NHL career, which put the Senators up by two with 8:32 remaining.
The Canadiens held a 20-11 advantage in shots in the first, though it was the Senators who took a 1-0 lead on Neil's second of the season midway through the opening period.
After failing to score on a flurry of chances following the opening faceoff, Montreal drew even late in the period when Cammalleri's shot from the right side struck Leclaire and floated past the Ottawa goalie and over the goal line.
Notes
Canadian Prime Minister Stephen Harper and Canadiens Hall of Famer Guy Lafleur helped unveil a set of commemorative postage stamps celebrating the team's 100th anniversary at a Bell Centre press conference prior to the game. … Kovalev, who has also played for the New York Rangers and Pittsburgh, had 103 goals and 161 assists for 264 points in 314 regular season games with Montreal from 2004-09. He added 17 goals and 14 assists in 33 playoff games with the Canadiens. … Cammalleri, who scored 39 goals with Calgary last season, signed a five-year, $30 million deal with Montreal on July 1.