Chandler Stephenson gave Vegas a 1-0 lead at 3:40 of the first period, scoring from the low slot after Evgenii Dadonov shot the puck toward the net from below the goal line.
Michael Pezzetta tied it 1-1 at 15:28 when he got the rebound of Artturi Lehkonen's initial shot.
Karlsson scored on the power play at 11:46 of the second to give the Golden Knights a 2-1 lead.
Tyler Toffoli scored a power-play goal to tie it 2-2 at 14:20, shooting from near the left post after receiving a pass from behind the net by Nick Suzuki.
Mike Hoffman gave Montreal a 3-2 lead at 6:15 of the third period with a one-timer short side from the left face-off dot.
"If you look at the last two games, we had the lead in the third period ... and tonight with the goal in the third with 13, 14 minutes (left), obviously on the other side you feel the urgency. They came back pushing again," Montreal coach Dominique Ducharme said. "It was some times where obviously they were storming us. Other times, we had better control of our game. I think we're trying to keep from the outside as much as possible to give less quality chances."
Marchessault tied it 3-3 at 11:39, scoring from the right circle off a cross-slot pass from Karlsson on the power play.
"I thought we put in the right amount of work to deserve two points tonight," Vegas coach Peter DeBoer said. "This league shows you don't always get what you deserve, but I thought we worked hard enough to get two points."