J.T. Compher had two goals and an assist, and Philipp Grubauer made 19 saves for the Avalanche, who will play their first Game 7 since 2014, when they lost 5-4 in overtime to the Minnesota Wild in the first round.
"It was a huge effort by our team tonight," Compher said. "We knew it was win or go home, and a bunch of guys stepped up tonight ... The plan was to force Game 7, and we were able to do that."
Brent Burns scored, and Martin Jones made 22 saves for the Sharks, who were 5-1 since 2016 with a chance to clinch a playoff series.
"We didn't play good enough to win this one," forward Timo Meier said. "We got to bring a better effort. It's a Game 7 now, we know want to do. It's going to be loud in our building, we're going to come out, and we're going to bring our best hockey."
Compher scored at 4:00 of the third period to give the Avalanche a 3-2 lead.