It was Demko's first NHL shutout.
"My job's the same regardless of the score," Demko said. "I just try to keep pucks out of the net. It doesn't matter if it's 1-0, we're down 1-0 or up by three. I'm just trying to make the saves I have to make to make sure they can find a way to win."
Quinn Hughes and J.T. Miller each had a goal and an assist for Vancouver, the No. 5 seed in the West.
"We got a belief in the room and lot of guys in here thought we had a good chance to win," Hughes said. "Sometimes guys are missing families in the bubble, but what I can say is after we win games, there's no better feeling. I think we're really coming together here."
WATCH: [All Golden Knights vs. Canucks Game 6 highlights]
The Canucks avoided elimination with a 2-1 victory in Game 5 on Tuesday. Teams that win Games 5 and 6 after trailing a series 3-1 are 29-28 (50.8 percent) winning a best-of-7 Stanley Cup Playoff series.
Game 7 is in Edmonton, the West hub city, on Friday (9 p.m. ET; NBCSN, CBC, SN, TVAS).
"I feel good, ready to go," Demko said. "It's a big Game 7. This is a game you want to play in, and as a kid you watch Game 7s, and you're envisioning yourself in those games, so I'm just really excited for tomorrow, and I know our group will be ready."
Robin Lehner made 19 saves for the Golden Knights, the No. 1 seed.
"There's no place for frustration right now," Vegas coach Peter DeBoer said. "We've got to win one game to move on. We'll reset, come out tomorrow, and we've put ourselves in a position here where we win a game, we get to move forward, and that's what we've got to concentrate on."
Demko, who made 42 saves in his first NHL postseason start in Game 5, is the first rookie goalie with a shutout in the playoffs for the Canucks. The 24-year-old made 17 saves in the second period and 22 in the third. He has made 66 consecutive saves since allowing a goal to Shea Theodore at 15:12 of the second period in Game 5.
"Thatcher's been our MVP [in Games 5 and 6]," Vancouver forward Jake Virtanen said. "He's standing on his head back there and making some amazing saves, keeping us in games. You can see the confidence he has back there. He's just calm and collected and playing his game."