Nathan MacKinnon had two assists, and Grubauer made 27 saves for the Avalanche (25-15-6), who have lost four straight games (0-2-2) and nine of their past 12 (3-6-3).
"I thought we were engaged start to finish, played hard, checked hard, worked hard defensively," Colorado coach Jared Bednar said. "I thought we played real hard, I don't know how else to summarize it. It seems like right now anything that can go wrong will go wrong, so we just got to keep grinding."
Denis Gurianov cut it to 2-1 at 9:54 of the second period, taking a pass from Roope Hintz in the right circle and scoring under the arm of Grubauer on the power play.
Jason Dickinson tied 2-2 at 14:37 of the third period, getting his own rebound and scoring with a backhand while falling down.
Nikita Zadorov gave Colorado a 1-0 lead at 9:04 of the first period when his pass to the slot redirected five-hole on Bishop off the stick of Stars defenseman Miro Heiskanen.
Gabriel Landeskog made it 2-0 at 15:27 on a deflection of MacKinnon's point shot.
"We knew that wasn't our best," Bishop said. "Obviously, we wanted to come out and look a little better, but they played a really good period and kind of threw it all at us. We knew that we had to be better, so it was kind of one of those things where you come in and regroup. It was only 2-0, we still had 40 minutes to play."
Alexander Radulov appeared to cut it to 2-1 with a power-play goal at 6:02 of the second, but Bednar challenged the play, and the call was reversed after a video review determined Jamie Benn was offside.
"I think we played really good the whole game," Grubauer said. "I don't think we gave up too many chances. We worked on the D-zone coverage a little bit and how we played off the rush. I think that was one of our better games.
"A little bit unlucky, we hit two posts in the second period that went out, they hit one in overtime and it went in, so that's how it goes. I think we can build on the last couple of games even though we lost. You've got to move forward."
