The Canadiens winger scored 4:31 into overtime and had two assists as Montreal downed the Pittsburgh Penguins 3-2 in the opening NHL pre-season game for both clubs on Monday night. "The first game is always the toughest - everything goes so fast," said Kovalev, who set up goals by Andrei Kostitsyn and Corey Locke.
Crosby scored in the second period after Kovalev was stripped of the puck by Maxime Talbot. The Penguins superstar also helped set up Ryan Malone's tying goal on a power play at 14:50 of the third period.
The teams meet again Tuesday night at the Bell Centre, although Crosby is not slated to play.
It will be tough to match the unusual intensity for an exhibition game that had the sellout crowd of 21,273 roaring at the end.
The teams were playing three skaters aside after Crosby and Mark Recchi erased their team's two-man advantage by taking roughing penalties when Kovalev cruised into the slot and beat Marc-Andre Fleury with a low shot.
That made up for his second-period gaffe, when he was stripped of the puck by Talbot to set up Crosby's goal.
"I'm kind of glad he made that mistake in the pre-season, but he came back hard after that," said Canadiens coach Guy Carbonneau.
The Canadiens lost two defencemen early to injury.
Andrew Archer went to hospital for x-rays on his face after a lopsided loss in a fight with Georges Laraque that left him bleeding heavily near the left eye. Carbonneau said the injury does not appear to be serious.
Mathieu Carle left with a knee injury that will need further examination on Tuesday.
Crosby was in the box for slashing Josh Gorges' stick and the Canadiens had a 4-on-3 advantage when Kostitsyn got the opening goal 6:41 into the second frame.
Kovalev's point shot was partially blocked and slid to Kostitsyn for a quick shot into an open side.
Crosby was penalized again for hooking Mikhail Grabovski at 14:49, but scored when he got out of the box on Talbot's set-up for a shot that went into the short side off Mike Komisarek's skate.
It was the first shot of the game for Yann Danis, who had replaced Cristobal Huet in the Montreal net.
Locke blasted a shot in from the top of the right faceoff circle 11:43 into the third period, but Malone tapped in Mark Recchi's pass from the crease on a play started by Crosby.
Fleury went all the way in goal for the Penguins.
Montreal rookie Kyle Chipchura's line with Tom Kostopoulos and Janne Lahti was given the task of checking Crosby's trio with Peter Sykora and Recchi and did a good job.
"I played against him in junior, but he wasn't that fast or strong then," said Chipchura, a strong candidate to start the season in Montreal. "Any time Crosby's in town, the place is full and there's lots of intensity.
"It makes it fun when you win."
It was the NHL debut for Penguins first-round draft pick Angelo Esposito, a Montreal native with a large following in the seats. Although he had a quiet game without a shot on goal, coach Michel Therrien said he played a good first game.
"I know he was nervous, but he didn't put us into any trouble," Therrien said.
The Penguins had 13 NHL regulars in their lineup while Montreal had 10.
A moment of silence was held before the game for former Canadiens player John Ferguson and former GM Sam Pollock, who both died of cancer over the summer.