Philipp Grubauer made 28 saves for the Avalanche, who won eight of their final 11 games of the regular season.
"The majority of this game I liked the way we played," Colorado captain Gabriel Landeskog said. "I liked the speed we played with. We were able to create some zone time, but at the same time, I'd like to get something in front of Smith and take away his eyes. He's a big goalie. He makes the saves he needs to. I'd like to test him a little bit more."
Game 2 of the best-of-7 series is Saturday at Calgary.
Mangiapane, a rookie in his first NHL playoff game, scored unassisted at 14:25 of the second period to put the Flames up 1-0. He got to a loose puck along the left wall in the Avalanche zone and cut to the net through the bottom of the left face-off circle before sliding in a backhand while on one knee.
Mangiapane's goal came after Smith made two glove saves earlier in the period during an Avalanche power play, on Landeskog's backhand from the edge of the crease at 2:13 and Derick Brassard's one-timer at 2:47.
"Margin of error and the difference isn't big this time of year, but he was definitely a difference-maker," Landeskog said of Smith. "Made some big saves when they needed him."
Tkachuk made it 2-0 at 18:58 when he tipped in Mark Giordano's point shot on the power play. Colorado coach Jared Bednar challenged for goaltender interference by Tkachuk, but the goal was upheld after video review.
Smith made a right-pad save on Alex Kerfoot in the slot on the power play at 6:27 of the third period, one second after Nathan MacKinnon hit the post on a wrist shot from above the right circle.
Smith made a glove save on MacKinnon's shot off the rush at 8:50.
Backlund made it 3-0 with a power-play goal at 17:01. His wrist shot from the top of the right circle went in off Grubauer's right skate through a Sam Bennett screen.
Smith got an assist on Tkachuk's empty-net goal to make it 4-0 with 2:45 remaining.
"He was awesome tonight, made a couple great glove saves, big pad saves," Tkachuk said. "He was great tonight. Playing the puck, giving me so much time on the empty-netter. Great play by him. Not many people see that. He was awesome tonight."