Darryl Sutter coached the Flames to victory in his first game behind Calgary’s bench after Geoff Ward was fired a week earlier.
Montreal’s record since Dominique Ducharme replaced Claude Julien on Feb. 24 is 3-2-3.
Derek Ryan and Sam Bennett assisted on both Leivo goals for Calgary (12-12-3). Jacob Markstrom stopped 17-of-18 shots for the win.
Corey Perry scored for Montreal (12-7-7) and Jake Allen made 27 saves in the loss.
The Flames were the more rested team having not played since Sunday. The Habs played their third road games in four nights, and were coming off a win in Vancouver the previous evening.
Calgary moved to four points back of Montreal for the final playoff spot in the all-Canadian North Division.
The Flames allowed a season-low 18 shots on net.
Perry tipped Shea Weber’s bobbling shot from the point between Markstrom’s legs at 10:15 of the third period to pull the visitors within one goal.
A Paul Byron slashing minor at 17:24 hampered Montreal’s comeback, however, as the Canadiens weren’t able to pull Allen for an extra attacker until just over half a minute remaining in the game.
Flames forward Milan Lucic and Habs winger Josh Anderson fought midway through the third period.
Weber’s pass from behind his own goal-line bounced off a teammate’s skate and into the open for Leivo to snare for his second goal of the game and third of the season.
The Innisfil, Ont., forward whipped the puck over Allen’s right shoulder at 13:55 of the second period for a 2-0 Flames lead.
Leivo scored his first at 1:52 finishing a tic-tac-toe passing play with Ryan and Bennett.
Sutter previously spent eight years in the Flames organization as coach and eventually GM between 2002 and 2010.
The 62-year-old from Viking, Alta., had last coached the Flames on April 17, 2006 in a playoff game.
Sutter also returned to the NHL’s head-coaching ranks Thursday after an absence of almost four years.
The Flame host the Habs again Saturday.
Notes: Montreal forward Tomas Tatar played his 600th career NHL game . . . The Flames beat the Canadiens at the Saddledome for the first time since March 9, 2017.
This report by The Canadian Press was first published March 11, 2021.