The Dockers led by 22 points at quarter-time and were still clinging to a slender margin as the clock wound down in Sunday's match in front of 43,882 fans.
Amartey gave Sydney the lead with 86 seconds remaining when he out-muscled Brennan Cox in a marking contest before calmly slotting the set shot from 20 metres.
In a frantic finish, Fremantle surged forward for one final thrust with seconds remaining.
A long kick into attack was marked by Dockers forward Jye Amiss about 25m from goal, but the siren had sounded just a second earlier, giving Sydney the 9.14 (68) to 9.11 (65) win.
The result marked Dean Cox's maiden AFL win as Sydney coach and ensured the Swans avoided slipping to an 0-3 record heading into their bye week.
Warner, who is being hotly pursued by Fremantle and West Coast, finished with 26 disposals, 10 inside-50s and 727m gained.
Sydney tagger James Jordon (24 disposals, 10 clearances) nullified the influence of Caleb Serong (19 disposals, five clearances, one goal) in a critical shut-down role.
Swans forward Amartey had hobbled off with a painful corked leg just minutes before reappearing to kick the match winner.
"To be able to fight through and find a way was really impressive," Cox said of Amartey, who finished with two goals.
"He was really sore towards the back end of the game, and to go back against their defenders ... and to give a really strong contest, and then to actually do it and convert, was really impressive."
Fremantle, who had a knack of losing close games last year, are 0-2 heading into next Sunday's western derby against the Eagles.
"I felt like we were a little bit maybe predictable the way we entered the ball inside our forward 50," Fremantle coach Justin Longmuir said.
"We had 31 inside 50s in the second half for four goals. So we need to make more of that.
"And then late we didn't quite execute the way wanted to execute. We need to be better at closing out games than we showed."
Jaeger O'Meara (32 possessions) and Nathan O'Driscoll (25) racked up big numbers for Fremantle, while Josh Treacy kicked four goals.
Star Dockers recruit Shai Bolton looked rusty on the way to 15 disposals and four clearances.
Fremantle defender Brandon Walker was subbed off in the second quarter after dislocating his left shoulder.
In a pressure-filled opening quarter, Fremantle made the most of their chances by kicking four goals to nil to take a 22-point lead into the first break.
Warner set up a goal and kicked one himself as Sydney fought back in the second quarter.
The second half was an arm wrestle throughout, but some critical mistakes from Fremantle in the dying minutes cost them victory.
