Kovalev, who played for Montreal from 2003-2009, opened the scoring late in the first. He added his second goal of the game on a power play 6:42 into the third, a slap shot through traffic past Carey Price that restored Ottawa's two-goal lead at 3-1.
Brian Elliott made 21 saves and Zack Smith also scored for the Senators, who have won six of eight since getting off to a 1-4-1 start. Erik Karlsson and Peter Regin each had two assists.
Brian Gionta scored with 4:07 left in the third to draw the Canadiens within one. Jaroslav Spacek also scored for Montreal, which won 3-2 in Buffalo on Friday. Price made 28 saves.
Kovalev got credit for his fourth goal of the season when he gave Ottawa a 1-0 lead 17:50 in. Karlsson's point shot through traffic struck the Russian veteran on its way past Price.
Smith, who was recalled from Binghamton of the AHL earlier in the day, put Ottawa up 2-0 in the second. He got his second of the season with a third effort on a loose puck Price was unable to cover up 12:15 into the middle period.
Spacek drew the Canadiens within 2-1 with 1:21 left in second. A video review confirmed that the Montreal defenseman's shot from the right point deflected off Sergei Gonchar's stick on its way past Elliott.
Montreal's Andrei Markov tipped a puck into the net 11:07 in. Referee Mike Leggo immediately waved off the goal because the Canadiens defenseman had made contact above the level of the crossbar.
NOTES: Kovalev has 417 goals and 580 assists in 1,242 games with the New York Rangers, Pittsburgh, Montreal and Ottawa. ... Elliott made his 10th straight start. ... Ds Alexandre Picard and Ryan O'Byrne and C Lars Eller did not dress for Montreal.