The Swans looked primed to pull off a comeback on Friday night after turning around a 27-point deficit to lead by a goal at the final break, but the Crows roared back to life amid bouts of heavy rain.
With the scores level at 10.8 (68) apiece in the fourth quarter, a desperate Luke Nankervis launched the visitors ahead en route to a 13.9 (87) to 10.11 (71) victory in front of 33,413 fans.
Learning their lessons after crashing out of finals in straight sets, Adelaide stamped their premiership credentials while dealing a blow to fellow flag fancies Sydney.
Last year's minor premiers, Nicks' men can finish the round as high as third after claiming a third straight win and condemning the second-placed Swans (13-5) to their first defeat at the SCG this season.
"I said to them post-game, I said I hope Adelaide's rocking at the moment because it would have been one of those ones," Crows coach Nicks said.
"I know my mum would have been sitting there stressing out and ready to abuse me anyway if we'd lost it, but she'd be up and about at the moment, which is good.
"There's a lot of belief here, so I'm mindful that we needed to have this.
"It's great that we've had this win. It reinforces our belief in ourselves."
Darcy Fogarty kicked four goals for Adelaide, but the star forward is sure to draw scrutiny after his high tackle on Harry Cunningham in the second quarter left his Swans opponent concussed.
Izak Rankine played a crucial role in the Adelaide engine room with one goal, 30 disposals and eight clearances.
He was well supported by James Peatling, while Mark Keane and Nankervis stood firm in defence.
Sydney have now lost their past three matches against top-six sides, following dismal defeats to Brisbane and Fremantle.
They remain second (13-5), with a four-point buffer against Adelaide.
"We've got work to do, as simple as I can be with that," Swans coach Dean Cox said.
Isaac Heeney shook off a lower-back issue to lead the home side with 29 disposals and four clearances.
The midfield superstar had sent a scare through the Swans camp in the third quarter, hobbling off sore after a contest for the ball with Crows forward Riley Thilthorpe.
Tom Papley also bagged four goals in his second game back from a calf injury, while Braeden Campbell kicked a major in his first AFL game of the year after a pre-season shin stress fracture.
Sydney started brightly with the first two goals, but it was all Adelaide in the opening term.
With 32 uncontested marks to Sydney's seven, the Crows punished the home side from turnovers to slam through five unanswered goals and take a 21-point lead at quarter-time.Â
Swans superstar Chad Warner got on the back of Nankervis to bring down a brilliant mark, only to cough up possession with a shoddy kick to Adelaide defender Keane.
Warner made amends after the break by hitting the scoreboard, before livewire Papley fired off back-to-back goals to help Sydney eat into Adelaide's game-high 27-point lead.
With eight points the difference at halftime, midfielder James Rowbottom mimicked Papley with two straight goals to give Sydney a six-point lead at the final break.
But it was Fogarty's turn to bag a double to kick off the final quarter, setting up a tense finish at the SCG.
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