The Swans trailed by eight points at three-quarter time, but prevailed 14.9 (93) to 13.12 (90) Adelaide Oval on Saturday night.
A week after coming from six points down at the last change to pip St Kilda, Sydney kicked five goals to three in the final term against Port.
The Swans (12 wins, two losses) join Fremantle (12 wins, one loss) at the top of the ladder, with the Dockers ahead on percentage.
"It was good to get away with it in the end," Swans coach Dean Cox said.
"It was a grind all the way through."
Sydney have a bye next weekend before a challenging month: Brisbane Lions (away), Western Bulldogs (home), Fremantle (away) and Adelaide (home).
Unfancied Port (15th, four wins, nine losses) held a match-high 20-point break midway through the second term.
The Power's initial half-dozen scoring shots were behinds and they led 3.6 to 3.2 at quarter-time.
Port kicked the opening three goals of the second term to create a 20-point break and held a 7.9 to 5.6 halftime advantage.
Their mid-season draftee Xavier Bamert, on AFL debut, featured in two flashpoints in the third term.
The 19-year-old, plucked from North Melbourne's VFL team, booted his first AFL goal eight minutes into the term.
But soon after, he gifted the Swans a goal: standing the mark, he misheard an umpire's call and advanced on Riley Bice, who was given a 50m penalty and goaled.
Bice's strike put his side two points down and star midfielder Isaac Heeney gave Sydney the lead minutes later with a classy long-range left-footer.
Heeney's strike was the fourth major in an ominous dozen minutes for the Swans, but Port responded with two rapid goals to take a nine-point buffer into three-quarter time, 10.11 to 9.8.
Flag fancies Sydney then booted four consecutive goals in the final term to turn their deficit into a 15-point lead.
But the Power again rallied with the next two majors to trail by three points with five minutes remaining.
Swan James Rowbottom crumbed a goal with two-and-a-half minutes left, only for Port's Jordon Sweet to convert with 34 seconds on the clock - the last score of the game.
"It was a really good performance from the team against quality opposition (but) we fall short again," Power coach Josh Carr said.
Sydney captain Callum Mills (29 disposals), Isaac Heeney (30, one goal), Chad Warner (26, one goal) and defender Nick Blakey (25) were prolific ball-winners, while Charlie Curnow and Logan McDonald booted three goals apiece.
Port's acting captain Zak Butters (27 touches, one goal) and Joe Richards (23, one goal) were standouts, while teammates Sweet, Mitch Georgiades and Corey Durdin slotted two majors each.
6°C