Gurianov tied it 2-2 at 1:25 of the third period when his one-timer deflected off the stick of Cole past Grubauer on the power play.
"You're going to play tight games all season and it's obviously a very good team over there," Bishop said. "It's a big win for us. You're going to have to find different ways to win every night."
The Stars had a chance to win in overtime after Nathan MacKinnon was called for tripping, but they were unable to get one of their four shots past Grubauer.
Joe Pavelski and Alexander Radulov scored in the shootout while Bishop stopped MacKinnon and Mikko Rantanen.
"We're playing better hockey," Pavelski said. "You can just feel it jumping over the wall. For us, it has to continue. It's one game. You have to build off of it now and we found a way to win. [Bishop] came up with some big saves and we got the extra point in the shootout."
Seguin scored 1:08 into the first period to give Dallas a 1-0 lead, tapping the puck into an open net after receiving a backhand pass from Radulov, who returned after missing the previous two games with a lower-body injury.
"That was a great first period," Stars coach Rick Bowness said. "I was a little concerned going into the game because they had played last night (lost 6-4 to the Minnesota Wild) and that worked to their benefit. I was very happy with that first period.
"That's against a very good hockey club. We found a way to get two points, and that's the most important thing."
Compher tied it 1-1 at 18:58 of the first when he got the rebound of Tyson Jost's shot in the crease.
Cole gave the Avalanche a 2-1 lead at 7:19 of the second period after jumping up to catch a drop pass that deflected off the stick of Miro Heiskanen, skating into the left circle, and scoring five-hole on Bishop.
"We were much more disciplined with the puck," Bednar said. "We checked hard. We gave up a couple chances that they earned. We had a couple at the other end. I thought it was pretty evenly played."
