Franklin kicked the last two goals of the game on Friday as Sydney won 12.10 (82) to 9.17 (71).
Sydney led by 24 early in the game and twice by 22 in the third quarter, but GWS nailed four straight goals either side of the final break to hit the front for the first time early in the last quarter.
Midfielder Luke Parker, who had 26 disposals and 11 marks levelled the scores with a goal and then Franklin put his imprint on the game.
The tenth highest goalkicker in VFL-AFL history piloted the ball through for successive goals to secure the win.
Held well earlier by old rival Phil Davis, Franklin finished with 19 disposals, eight marks and kicked 2.2
Parker, Callum Mills Isaac Heeney and Nic Newman were strong performers for Sydney.
Kieren Jack kicked three majors for Sydney, with Franklin, Sam Reid and Parker booting two apiece..
Lachie Whitfield and Callan Ward sparked the Giants with 29 and 24 touches respectively, driving them forward at every opportunity.
Jeremy Cameron kicked three goals for GWS and pulled down seven marks.
Toby Green showed some nice touches in his first game of the year, but Brett Deledio had a quiet season debut, notching just one kick and eight touches in total.
The Swans kicked the first four goals and led by 17 at the first change.
GWS fought back in the second quarter, kicking three goals to one, to trail by one at halftime, and were down by two at the last change.
© AAP 2018
SYDNEY 4.3 5.4 9.8 12.10 (82)
GREATER WESTERN SYDNEY 1.4 4.9 8.12 9.17 (71)
Sydney: K Jack 3 L Franklin 2 L Parker 2 S Reid 2 H Marsh J Kennedy W Hayward.
Greater Western Sydney: J Cameron 3 D Lloyd H Himmelberg J Patton L Whitfield M De Boer T Greene.
Sydney: L Parker C Mills I Heeney N Newman H Cunningham L Franklin.
Greater Western Sydney: L Whitfield C Ward J Cameron T Taranto D Simpson S Coniglio.
Greater Western Sydney: L Tiziani (AC joint).
Venue: Blacktown Olympic Park.