Four goals down at one stage, Braehead came back from the brink to claim a 6-4 lead, before eventually being forced into overtime to eventually claim their first win of the weekend.
It tops off a successful first term for head coach Ryan Finnerty in his maiden season at the Braehead Arena, with the Glasgow club reaching the play-off finals weekend for the first time in their history under his stewardship.
Finnerty rested first choice netminder Kyle Jones for the match with Fife, who went down 1-0 in their semi-final to Belfast Giants, and replaced him with long-serving understudy Mike Will.
It was to turn out to be a busy afternoon for the Englishman, who had to pick the puck out his net twice in the first period.
Firstly Euan Forsyth picked up a pass at the blue line and sent it rocketing high into the goal after just under five minutes, with Josh Scoon doubling Fife's advantage after 15:09.
It could have been worse just moments later, but Will stood up well as the imposing figure of Matt Nickerson raining down one on one failed to make the keeper blink first.
Clan's frustrations were not confined to defence, however, as several opportunities were squandered at the other end. Kevin Bergin, Joel Champagne and Lee Esders were the main culprits, while a lot of huffing and puffing in the opposing third failed to amount to a goal.
Given Braehead had their fraught encounter with Sheffield last night before then playing first this afternoon, it was always going to be a physically tough ask on the Glasgow club's players to be at 100%.
And it took just 13 seconds for matters to get worse. Flyers' Jamie Wilson took two attempts at the crease to hack the puck over the line from a Stephen Gunn pass for three.
Three minutes and 20 seconds later it was four; Scoon skating in unchallenged through the centre before dropping his shoulder and gently sliding the puck under Will with Clan seemingly dead and buried.
However, it was with to trigger an extraordinary response from Finnerty's team, who went on to score six in a row to roar into a stunning lead.
First Tristan Harper turned in a Joel Champagne rebound to start the run with nine minutes left in the second, with Shane Lust rocketing a fierce shot high beyond Blair Daly five minutes later to make it two. Clan still had time to find a third before the period was over, Kenton Smith this time thumping into the goal.
The third period started in a similar fashion to the way the second had finished, with Braehead scoring. Just under two minutes had elapsed when Champagne poked home from close range to level.
Fifty nine seconds later they took the lead through Matt Haywood after a squared Kevin Bergin pass opened up a chance for the Englishman.
A second for Champagne with just 16 minutes left to play looked to put the game beyond the Fife side. Yet late counters from Derek Roehl - who netted after going through one on one with Will - and Ned Lukacevic sent the tie to four-on-four overtime.
But Clan were not to be denied third place with Ed McGrane popping up on the left flank, skated in towards goal and guided the puck into the net.
Ryan Finnerty, Clan coach: "It's a difficult game for the guys to get up for after such a short turnaround from last night. It was a good game and I'm happy to end the season on a win.
"It's hard when there's not a whole lot on the line but they were professional. You don't want guys getting injured at this stage of the season.
"But I thought it was a decent game, at least it went to overtime and gave the fans some goals."
Todd Dutiaume, Flyers coach: "I watched Braehead's game after ours on Saturday night and it's fair to say both teams left everything on the ice.
"Guys were obviously disappointed, but we picked ourselves up. A few of the guys' competitive spirits came out in the end."