Dube cut it to 2-1 at 15:06 of the second period when he one-timed a pass from Nikita Zadorov, beating Saros five-hole.
"I don't think we got down on ourselves," Dube said. "Even that first period we had some good momentum, gave them a couple chances. I think we stuck to it. We played hard in the second period. Guys were on top of the puck playing hard."
Hanifin tied it 2-2 at 4:45 of the third period with a wrist shot from the top of the left circle through a screen that beat Saros glove side, and Coleman put Calgary up 3-2 at 6:24 when he kept on a 2-on-1 and shot short side over Saros' glove.
Nazem Kadri scored into an empty net at 18:21 for the 4-2 final.
"Everyone was skating," Coleman said. "We didn't give them much in the third. That's the way we need to play if we're going to continue to win games. It wasn't one guy. It was a collective effort and that's what our team's got to be."