Evgeni Malkin gave the Penguins a 3-2 lead with 4:17 remaining. He scored from the right circle through a screen off a pass from Dominik Kahun, who had three assists.
Matt Murray made 28 saves for Pittsburgh (27-12-5), which is 6-1-1 in its past eight games.
"It was another really gritty, hard win, and that's what we needed tonight," Murray said. "That's a really good team over there, so we just stayed resilient no matter what happened."
Nathan MacKinnon and Gabriel Landeskog scored for the Avalanche (25-15-5), who are 0-2-1 in their past three games. Francouz made 26 saves in the first of a five-game homestand for Colorado.
"I think it's great for us that we were able to bounce back and tie it with the empty net," Francouz said. "It's a good point, but I think we deserved two points."
MacKinnon scored on the power play to give the Avalanche a 1-0 lead at 14:38 of the first period. Samuel Girard passed to MacKinnon in the left circle for a one-timer inside the near post.
Blueger tied it 1-1 at 13:29 of the second. John Marino's shot from the right circle deflected off Kahun to Blueger to the left of the net for a shot that beat Francouz short side six seconds after a Pittsburgh power play expired.
Landeskog made it 2-1 at 18:20 of the second when he lofted the puck from outside the Penguins blue line and it bounced past Murray's left arm into the net.
"That's a tough read, I can't really expect that kind of bounce," Murray said. "I'm just trying to get out and play and settle it down so our defense can make the play, and it bounced sideways on me. I just brushed it off as quickly as I could."
Bryan Rust tied it 2-2 for the Penguins at 1:14 of the third. After Francouz made the save against Malkin, Rust got the rebound, circled above the right hash marks, and scored with a five-hole shot.
"You give any team in this league just one opportunity, they are going to take advantage of it," Makar said. "It's unfortunate for us they did tonight. I mean, it's just a learning curve for us. We can't give that up in the third period, and for us, it's definitely not a confidence builder when we drop games like that."
A goal by Malkin at 16:34 of the first period was waved off because Rust interfered with Francouz.