Monahan finished with two goals and Matthew Tkachuk also scored as Calgary got its fifth win this season when trailing after two periods. Anaheim (two) is the only other NHL team with more than one win when behind after 40 minutes.
David Rittich had 12 saves to help the Flames win their fourth straight.
Jonathan Toews got his 300th career goal, and Jan Rutta and Brandon Saad also scored for the Blackhawks, who went 0-3-0 on a three-game trip to Western Canada. Corey Crawford stopped 36 shots, including 18 of the 20 he faced in the third period.
With Calgary trailing 3-1, Tkachuk started the comeback with 1:34 left in the second.
After pressing all period, Monahan finally tied it with 6:01 left in the third, when he neatly converted a cross-ice pass off the rush from Johnny Gaudreau. Monahan extended his point streak to six games (four goals, six assists).
Frolik put Calgary ahead for good with 4:55 left as his centering pass from below the goal line deflected off a player and got past Crawford. It was the seventh goal for Frolik, who scored only 10 times last year.
Backlund capped the scoring with an empty-netter with 59 seconds left.
Calgary opened the scoring 4:46 in when a quick passing sequence was finished off by Monahan in the slot on the power play.
Chicago tied it with 6:03 remaining in the period after Travis Hamonic's holding penalty put the Blackhawks on a two-man power play fpr 1:10 and they took advantage on Toews' eighth goal.
The Blackhawks took their first lead with 27 seconds left in the first as Rutta faked a slap shot to elude the sliding Garnet Hathaway, then put a wrist shot through a crowd and past Rittich high on his blocker side.
Saad split the Flames' defense while being harassed from behind by Mark Giordano and sent a shot that squeaked through Rittich's pads to make it 3-1 with 6:19 left in the second.
Calgary took over after that.
The Flames got an early five-minute man advantage when Duncan Keith was given a boarding major and game misconduct just over two minutes in, when his hit from behind on Dillon Dube sent the rookie head first into the boards. Dube left the ice on his own, but very slowly. He did not return. ... Chicago lost in regulation for the first time in its last 10 trips to Calgary (5-1-4).
Blackhawks: Host Carolina on Thursday night.
Flames: At Anaheim on Wednesday night to open a three-game California trip.