Marchessault tied it 1-1 at 3:13 from the right circle.
William Carrier gave Vegas a 2-1 lead at 7:23 when he deflected a Nicolas Roy shot.
Sillinger tied it 2-2 at 11:53 with a one-timer off a pass from Bjorkstrand, and Bjorkstrand put Columbus in front 3-2 at 12:23 from the high slot.
Sillinger tipped a Bjorkstrand shot to make it 4-2 at 5:36 of the second.
"I just kind of went to put my stick out and it hit my stick there," Sillinger said. "So fortunate for a lucky bounce."
Sillinger completed the hat trick and made it 5-2 at 7:26, converting a pass from linemate Domi on the rush.
"Obviously, the Sillinger line there, they were feeling it tonight," Blue Jackets coach Brad Larsen said. "The guys just keep pushing, they just keep pushing and playing hard. You know that's a desperate team we were playing. We knew where they were at. We talked about it. They need these points bad, so you better be focused and ready."
With center Boone Jenner (back) and forward Jakub Voracek (knee) out after each sustained an injury during a 3-2 shootout win against the Minnesota Wild on Friday, Sillinger was given a chance to earn more minutes. He played 16:58, his most ice time in 12 games.
"He's got a tremendous upside on his offensive game," Larsen said. "We always forget he's 18. At least, I think some people do. And he's going through the grind of a season. He's a kid that sees an opportunity. You know, some guys when they see these opportunities, they get excited. And it's amazing what some guys can do. Some guys freeze. He doesn't seem to be that kid."
Sillinger became the fifth Blue Jackets rookie to score a hat trick and the first since Pierre-Luc Dubois in 2017-18.
"That's special," Domi said of Sillinger. "He's such a good kid. He's so humble, he works so hard."
Evgenii Dadonov pulled the Golden Knights within 5-3 at 8:28 before Jack Eichel cut it to 5-4 on the power play at 2:40 of the third period.