And in Sunday's other matches, Brisbane forward Dakota Davidson kicked an equal career-high four goals to lead Brisbane to a crushing 48-point win over Essendon, while Fremantle were held goalless in the first half on the way to a 21-point loss to Port Adelaide in Perth.
A star display from veteran Ebony Marinoff (27 disposals, four clearances, 500m gained) put Adelaide ahead by two goals entering the final quarter of their round-one encounter.
But the Swans, under new coach Colin O'Riordan, never gave up.
Cynthia Hamilton kicked a goal early in the final term to put the Swans within touch, and Holly Cooper kicked her second goal of the match a short time later to put Sydney in front.
The margin sat at just two points for most of the rest of the term before debutant Imogen Brown kicked the winning goal in the final minute, sealing the 9.7 (61) to 7.11 (53) victory for Sydney, who missed the finals on percentage last year.
Smith, a two-time premiership winner at Brisbane, starred in her first game at the Swans with three goals and five marks, while Hamilton and Cooper kicked two goals apiece.
Swans speedster Zippy Fish, who finished with 23 disposals, eight tackles and 438m gained, said O'Riordan gave Sydney belief they could snatch victory during his three-quarter time address.
"We went into one of our pre-season games and we were down, so he pretty much gave us the same messaging from that game, and we came back and won that game," Fish said.
"(He told us) just have a bit of belief and be brave with the ball."
Christina Leuzzi kicked two goals for the Crows, while Madison Newman was prominent with 24 disposals, 443m gained and one goal.
Over in Melbourne, Brisbane played like a team possessed in the first half, storming to a 35-0 lead by early in the second quarter before 75-game milestone star Bonnie Toogood finally kicked Essendon's first goal.
The Bombers were able to largely match it with Brisbane from that point onwards, but the damage had already been done.
Davidson finished with 4.2 and five marks in a strong display in attack, while Catherine Svarc starred with 21 disposals and two goals in the 12.13 (85) to 6.1 (37) win.
Essendon were well led by Madison Prespakis (30 disposals, five clearances, 472m gained) and Maddison Gay (26 disposals, 523m gained), but the Bombers were simply blown away by Brisbane's early onslaught, which included five goals to nil in the opening quarter.
Fremantle didn't score their first goal of Sunday's match until former skipper Hayley Miller kicked truly at the eight-minute mark of the third quarter.
Port had already opened up a 27-4 lead before that goal went through, with the Power able to limit Fremantle to just two goals for the match in a comfortable 6.5 (41) to 2.8 (20) win.
Power ruck Matilda Scholz (25 hitouts, 16 possessions) helped her team win the clearance battle 27-16, with Ashleigh Woodland (21 disposals, 482m gained, one goal) and Abbey Dowrick (23 disposals, 503m gained, one goal) snaring seven clearances apiece for the Power.
Kiara Bowers (24 disposals, five clearances, nine tackles) battled hard for Fremantle, but the Dockers lacked fluency and composure under the intense tackling pressure from Port.
12°C