Ryan Johansen, Craig Smith, Kevin Fiala, Zac Rinaldo and Filip Forsberg did the damage for Nashville.
Just 51 seconds in, Johansen banged a puck behind the net from the halfboards - also knocking TJ Brodie's stick out of his hand - and then he cut hard to the net all alone and took a pass from Forsberg, firing the puck far-side over Smith's glove.
From there, the Flames rallied to tie the game on three separate occasions, but were unable to grab the lead at any point in the game.
"From the first minute, we were chasing the game, and every time we tied it up, we did something to get in our own way," said head coach Bill Peters. "We weren't as good as we needed to be.
"I thought they were good, they forechecked well and didn't give us much room. Once we got going, I thought we did the same thing. But every time it looked like we were going to take a step and get some momentum, we gave it back."
After dancing up the ice to gain the zone and get the puck deep, Gaudreau eventually got the puck back behind the net off a pass from a pinching Giordano and quickly wired it right onto the tape of Lindholm, who was out front parked to the right of Rinne, and he snapped it home with 1:17 left in the opening frame.
It was just Calgary's second shot of the period, with Nashville holding an 8-3 edge in that category after 20 minutes.
The Preds restored their lead after a prolonged possession in the Flames zone that ended with Smith getting the puck all alone in front of the net and he pulled the puck to his backhand and squeezed the puck shortside past the Calgary netminder at the 4:09 mark.
With Viktor Arvidsson in the bin for slashing, the Flames made good on a powerplay chance when Tkachuk scored his third goal of the season, tipping Lindholm's shot from the right flank around Rinne at 12:51.
Giordano picked up his second assist of the night on the marker.
"They're a good team. They play patient," the skipper said. "When they didn't have a sustained forecheck, they sat back in the neutral zone and waited for us to make a mistake.
"As the game went on, I thought we got better at it, but we have to do a better job of getting bodies and pucks, getting them through and pounding some rebounds home."
It didn't take long for the Preds to go ahead again, when Fiala scored a powerplay marker just 1:21 later.
The game was knotted up again by Bennett 2:20 into the third, when Derek Ryan threaded a beautiful cross-ice pass on the rush past two defenders to his linemate, who tucked a backhander into a yawning cage as Rinne slide across the opposite way and ended up with Fiala crashing into him.
He was slow to get up and looked a bit shaken.
A few minutes later, Rinne left the game and backup Juuse Saros went between the pipes with 15 minutes to play.
"It's unfortunate," Bennett said. "We started to pick it up again in the second and third and had a good pushback, but they capitalized and we didn't."
Calgary had some great chances to tie it up, including a great chance by Bennett with a tip at the side of the net that Saros was able to get enough of to stop.
"I thought our line generated a lot of chances, but I'm going to have some nightmares about that one I missed," Bennett said.
"That's way it goes."
Rinaldo scored the game-winner on a two-on-one at 6:31, skating the puck up the ice and then firing a low shot that beat Smith.
Forsberg added an empty-netter with 31.5 seconds to play.
"We didn't generate enough offence at the start," said James Neal, who had one of his best games as a Flame with a team-high four shots in just over 14 minutes of ice time.
"We were in the game and we had a chance to win it in the third. Tied it up, but a couple tough bounces.
"We'll look at it and get better."
