The Power had snatched the lead less than two minutes earlier on Saturday night when forward Jeremy Finlayson slotted a set shot with a snap on the way to a 9.12 (66) to 9.10 (64) victory.
Finlayson finished with a game-high three goals as the Power kicked four of the last five majors in a remarkable turnaround after trailing by as much as 20 points early in the final term.
The Swans looked to be cruising to a brave win halfway through the final term despite having their defensive pillars Paddy and Tom McCartin both subbed out before halftime due to head knocks.
But goals to Todd Marshall, Connor Rozee, Jed McEntee and then Finlayson shook up the pulsating contest and gave the Power an important second win of the season.
Zac Butters (27 disposals, one goal) and Xavier Duursma (22) battled hard for the Power, while Rozee overcame a slow start to finish with 20 touches and a crucial goal.
Aliir Aliir got the first hand to Florent's last gasp shot, and finished with 16 disposals, including 12 intercepts ,while restricting Lance Franklin to only eight touches and one behind in a pivotal duel.
"He likes playing at the SCG," Power coach Ken Hinkley said with a smile.
"He's been fabulous for us, without being too silly.
"It's always a challenge for players playing against their old club, Aliir has got great friends here, but it's a great opportunity for him to come here and show he likes his new environment."
It was a cruel outcome for Florent who had been outstanding for the Swans with 33 disposals and an earlier goal, while Chad Warner had also led the resistance when the Power first surged either side of halftime.
Warner finished with 30 touches (19 contested) and two goals, while Luke Parker (22) and Callum Mills (21) were also influential.
The Swans (2-2) will now be sweating on the health of Paddy McCartin after he knocked his head on the ground, then had to be helped from the field and subbed out early in the opening term.
The former No.1 draft pick has revived his career since joining the Swans as a key defender last year, after his five seasons with St Kilda were marred by eight concussions.
"I just spoke to him and he's OK," Swans coach John Longmire said.
"We just removed him, we made a decision. He didn't get a test or an assessment, we just removed him from the game."
Paddy's younger brother Tom was concussed when diving for a loose ball and colliding with a Power opponent just before the main break to leave the Swans without their defensive pillars as they eventually ran out of legs.
Peter Ladhams booted the opening goal and played through the match without major incident after being suspended for a gut punch on Ollie Wines the first time he faced his former side last year.
Both teams head to South Australia next for the inaugural Gather Round with Sydney facing Richmond on Friday night, while Port Adelaide host Western Bulldogs on Saturday night.
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