Patric Hornqvist scored with 1:12 left in the third period, and Jarry made 32 saves in his first start of the season for the Penguins (8-5-2).
Pittsburgh, which won 3-2 at the Edmonton Oilers on Wednesday, is 0-4-1 and has been outscored 31-8 in the second of back-to-back games this season.
"This is one of the better games that we played, back-to-back or no back-to-back," Penguins coach Mike Sullivan said. "This is definitely the best game we've played on this trip. We probably deserved a better fate. We liked a lot of our game tonight."
Monahan scored when he centered a rebound on Johnny Gaudreau's shot off Penguins defenseman Brian Dumoulin and into the net to give the Flames a 1-0 lead at 15:53 of the third period.
Hornqvist tied the game 1-1 when he deflected Kris Letang's point shot past Smith. Jarry was on the bench for the extra attacker.
"To be able to tie it up was good," Penguins captain Sidney Crosby said. "Unfortunately, we couldn't get it done in overtime."
Smith made 19 saves in the first period, including a glove save on Hornqvist's shot from a sharp angle on the power play at 10:23, and a left pad save on Carl Hagelin's partial breakaway at 12:38.