Sillinger redirected a shot from the blue line by Yegor Chinakhov.
"The puck came around and I tried to find that hole and get to the net," Sillinger said. "It deflected off my stick and went in the net."
Elvis Merzlikins made 34 saves for the Blue Jackets (7-3-0), who have won three straight despite trailing in the third in each of those games.
"I love how we came out in the third. We came to win a game," Columbus coach Brad Larsen said. "That's a good sign. They wanted to go get that win. That's a great sign for a young hockey team."
Andre Burakovsky scored twice, and Darcy Kuemper made 38 saves for the Avalanche (4-5-1), who also lost 5-4 in overtime to Columbus on Wednesday.
"The third was probably one of the worst periods we've ever played," Colorado forward Mikko Rantanen said. "In the third, we didn't have our legs on either offense or defense.
"They tied the game, and we were slow defensively, not physical enough. Sometimes it looked like we were playing just effortless."