Kovalev set up Richard Zednik's tiebreaking goal midway through the third period as the Canadiens capped a comeback unprecedented in their rich history, beating the Bruins 2-0 Monday night to win the first-round series in seven games.
"When we lost the first game and were down 3-1 in the series, we never gave up," Kovalev said.
For the second time in three years, the Bruins were upset by the Canadiens in the first round.
"We had our chances," Boston defenseman Nick Boynton said. "We blew it."
The Canadiens won for the first time after trailing 3-1, capturing the final three games, two of them in Boston. Seventh-seeded Montreal will take on the Eastern Conference's top team, the Tampa Bay Lightning, in the second round.
Jose Theodore, who helped eighth-seeded Montreal eliminate top-seeded Boston in the first round in 2002, got the first playoff shutout of his career, stopping 32 shots with the help of a swarming defense.
"Teams are successful because of goaltending in this league," Canadiens coach Claude Julien said. "He was at his best when it really counted."
Zednik clinched the victory with an empty-net goal with 7.8 seconds remaining. Kovalev and Saku Koivu assisted on both goals.
Kovalev leads the NHL with five playoff goals this season but allowed Glen Murray to go in by himself for the winning overtime goal in Game 4. That followed Kovalev's disappointing season with the New York Rangers and Montreal. In 12 regular-season games with the Canadiens, he had one goal and two assists.
"The most important thing was to save it for the playoffs," he said.
Bruins captain Joe Thornton played all seven games in pain, caused by a rib injury sustained with two games left in the regular season. He went scoreless in the series, and when it was over coach Mike Sullivan revealed the nature of the "upper body" injury.
"Most players in the league wouldn't have played," Sullivan said.
But Thornton was determined.
"It's something you have to do in the playoffs," he said. "It's devastating [that] we lost the game. It's unacceptable."
Sullivan said his team played a "terrific" game. But a goal that Boston goalie Andrew Raycroft could do little about was costly.
Kovalev skated behind the Bruins' goal and shot the puck from the goal line to Raycroft's right. It hit the side of the net and bounced in front, where Zednik was unguarded. He scored with 9:08 left.
"On the shift before the goal, I told Zednik to get in front of the net and, with the choppy ice, I'd try to get it in front," Kovalev said.
Boston, which had never lost a series it led 3-1, had its six-game winning streak in Game 7s snapped. Montreal has won four of the six Game 7s between the Original Six teams.
The Canadiens were 0-12 in seven-game series they trailed 3-1. They got as close as 3-3 just one other time, in 1954. The Bruins were 17-0 when ahead 3-1.
Raycroft was sharp for most of Game 7, especially in the second period, when the more aggressive Canadiens held a 15-9 shots advantage.
But the Bruins showed renewed spark in the third period, constantly attacking and then getting back to stop Montreal's rare threats.
Finally, Zednik scored on the Canadiens' first shot of the period.
One of the Bruins' few scoring opportunities in the first two periods came with 6:45 gone in the second, when Sergei Samsonov took a 20-foot shot from the left that Theodore stopped with his pad. Mike Knuble got two rebound shots after the save but was turned away.
"I'm sure there were a lot of people who had doubts after Game 4," Koivu said. "We had concerns, too, but we were able to refocus."
The last time the Bruins lost a Game 7 was April 25, 1982, against Quebec. ... Montreal's last Game 7 victory was May 1, 1992, when the Canadiens eliminated Hartford. ... The Bruins were shut out 11 times this season, nine times at home. ... Injured Boston Red Sox shortstop Nomar Garciaparra and retired baseball player David Justice attended the game.