Allen stopped three breakaways and made a difficult save against Colin Wilson at 12:16 of the third period that left the Avalanche forward shaking his head.
"It was a good game, that was hockey," Allen said. "Two hard-playing teams going at it. We're desperate and they're on a roll. We're trying to climb our way back and they're trying to stay on top, and it's a fun, entertaining game."
Mikko Rantanen and Nikita Zadorov scored, and Varlamov made 34 saves for the Avalanche (15-6-5), who are 8-0-2 in their past 10 games.
"It's a big point, it's an important point," Avalanche coach Jared Bednar said. "We would've liked to have two, but sometimes that's the way it goes."