The teams were three aside after Sidney Crosby and Mark Recchi erased a two-man advantage by taking roughing penalties when Kovalev cruised into the slot and beat Marc-Andre Fleury with a low shot.
Crosby scored in the second period and helped set up Ryan Malone's tying goal on a power play at 14:50 of the third for Pittsburgh.
Andrei Kostitsyn scored and assisted on Corey Locke's third-period goal for Montreal.
The teams meet again Tuesday night at the Bell Centre, although Crosby is not slated to play.
The Canadiens lost two defencemen 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. And Mathieu Carle left with a knee injury.
Crosby was in the box for slashing Josh Gorges' stick - less than a minute after Andrei Markov was sent off for tapping Crosby's 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. Maxime Talbot stripped Kovalev of the puck and set up last season's NHL scoring leader 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 fine job.
The Penguins had 13 NHL regulars in their lineup while Montreal had 10.
A moment of silence was held before the game for former player John Ferguson and former GM Sam Pollock, who both died of cancer over the summer.