Fremantle had to fight back from 26 points down late in the third quarter to eke out the terrific 11.14 (80) to 12.7 (79) win on Saturday night at Optus Stadium.
They took a six-point lead with three minutes remaining in the blockbuster in front of 57,507 fans before a dramatic finish saw Suns defender Mac Andrew drift forward, take an important mark to goal and level the scores with 72 seconds to go.
Then the veteran Swallow, who was subbed on early in the final quarter, received a free kick after being tackled by Brennan Cox without the ball.
Swallow nailed the 50m set shot for what at first appeared a goal. But on review it was only awarded a behind after replays showed it was touched on the mark.
That actually ended up being good for Gold Coast.
Rather than the centre bounce that a goal would have triggered, it left Fremantle with just nine seconds to go coast-to-coast and the Suns held firm to secure the narrowest of wins.
Gold Coast will now take on Brisbane in a Q-Clash semi-final at the Gabba next Saturday night.
"We were just hoping he made the distance, to be honest. We weren't too sure," Gold Coast coach Damien Hardwick said with a laugh when asked about Swallow's winning set shot.
"It was sort of a fairytale moment, wasn't it?
"He is Gold Coast. He is the fabric of the place, and what he's given our football club - it was just reward."
The result ends the decorated career of Fremantle's dual Brownlow medallist Nat Fyfe, who received a guard of honour while being chaired off by Alex Pearce and Luke Ryan after the match.
"It's devastating," Fremantle coach Justin Longmuir said.
"There's no pleasant way to go out, unless you win it.
"But the way that game played out, and the fact Nat's retiring and Sonny (Michael Walters) is retiring, it adds a lot of extra emotion.
"So to stand up in front of the whole staff and group after that performance and that result ending with those two greats retiring, it's really hard to put into words.
"The players are hurting."
Bailey Humphrey kicked three goals for the Suns, while Matt Rowell (34 disposals) and Noah Anderson (32 disposals) were huge.
Caleb Serong (35 disposals) and Andrew Brayshaw (32 disposals) racked up big numbers for Fremantle and Patrick Voss kicked three goals.
Fremantle entered the match with a combined 51 games of finals experience to their name.
In contrast, the Suns had just three players with finals experience - Daniel Rioli (13 games), John Noble (seven) and Ben Long (one), but it was Fremantle who looked more nervous in the first half.
Just one goal apiece was kicked in an even opening quarter as both sides blew a series of golden chances in the slippery conditions.
The goals flowed more freely in the second term as the game opened up, with Humphrey kicking three of Gold Coast's seven majors.
Fremantle initially kept pace with the Suns - the home side holding a one-point lead when Murphy Reid nailed his set shot at the 18-minute mark.
But four quick goals from the Suns to end the half turned the contest on its head, with Gold Coast entering the main break with a 25-point advantage.
The Suns toyed with Fremantle for most of the third quarter, but their inability to capitalise on their territory dominance came back to haunt them.
Dockers goalsneak Isaiah Dudley, who had tallied just two disposals in the first half, kicked two goals in the space of three minutes late in the quarter to ignite the crowd and reduce the margin to 15 points at the final change.
In a frantic final term, Luke Jackson took a huge mark over Mac Andrew with less than four minutes remaining to kick the go-ahead goal, giving Fremantle a six-point lead.
But the Suns had the last laugh, coming up trumps in the clutch moments at the clock wound down.
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