Monahan and Tkachuk each scored in the shootout, and Rittich stopped Huberdeau and Barkov.
"You want to squeeze out every point you can, and when you come back the way we did from a 4-2 deficit, you try to find the positive in that," Florida defenseman Aaron Ekblad said. "We're getting points on the road, we've got to keep putting them in the bank and try to get a couple wins here to make it a successful road trip."
Tkachuk redirected Michael Frolik's pass on a 2-on-1 to give Calgary a 3-2 lead 32 seconds into the third period, and Giordano made it 4-2 at 5:16.
Josh Brown scored his first goal of the season on a slap shot through traffic to cut Calgary's lead to 4-3 at 6:20, and Evgenii Dadonov tied it 4-4 at 8:41 when he tapped in Huberdeau's pass on a 2-on-1.
"It just seemed like when it was 4-2 they were pressing and we had no answer for them," Tkachuk said. "Maybe a little sloppy with our puck play, and stuff like that. You learn from it. For our team to be successful we're not going to win those 6-5 [games], trying to come back after being [down] … that's a stressful game."
Mark Pysyk put Florida up 5-4 at 14:59, and Bennett tied it 5-5 at 17:17.
"It's a high-scoring game so it's obviously tough for goalies," Bobrovsky said. "Lots of scoring chances, not many shots, but all of them pretty much scoring chances."
Czarnik scored on the power play to put Calgary up 1-0 at 11:43 of the first period when his pass to Rasmus Andersson deflected in off Ekblad's skate.
Huberdeau tied it 1-1 4:28 into the second period. TJ Brodie turned the puck over in the neutral zone before Huberdeau shot over Rittich's shoulder off the rush.
Tkachuk made it 2-1 when he took a return pass from Frolik and shot short side at 11:59.
"I think in big games [Tkachuk] is always going to be there," Giordano said. "You always want the puck on his stick, it seems, in high-pressure situations. He makes plays under pressure. That's what makes him so good, no different from tonight. He made a couple great plays offensively that helped us score goals."
MacKenzie Weegar tied it 2-2 with a wrist shot from the high slot at 15:15.