After Anisimov put the Blackhawks ahead 3-2 at 7:14 of the third period, Marchessault scored 41 seconds later to tie it 3-3 at 7:55, and Tuch scored at 8:07 to put the Golden Knights ahead 4-3.
"We don't panic," Karlsson said. "When they scored the 3-2 goal, we just tried to keep going. That's what we did. It's been the story so far that we just keep going, and that's why we've been successful lately."
Smith scored on the power play at 1:28 of the first period to give Vegas a 1-0 lead.
Karlsson got his 150th NHL point, 100th for the Golden Knights, with an assist on Smith's goal.
Toews cut the Vegas lead to 2-1 four minutes into the second period, and Strome tied it 2-2 at 16:15 after Patrick Kane's shot caromed off the boards and bounced to him at the edge of the left face-off circle.
The Golden Knights appeared to take a 3-2 lead at 17:45 of the second when forward Oscar Lindberg's shot crossed the goal line prior to the Blackhawks net being dislodged. But the goal was waved off for goaltender interference, and the call on the ice was upheld by video review after Gallant challenged it.
Chicago coach Jeremy Colliton challenged for goaltender interference by Max Pacioretty on Tuch's goal, but it was upheld after video review showed Blackhawks defenseman Brandon Manning caused Pacioretty to contact Crawford.
"Every win is important. Every game is important," Tuch said. "We have to come in and play 60 minutes every single night."