Heeney, who has tormented the Kangaroos in recent times, kicked three vital goals in the Swans' hard-fought 16.9 (105) to 14.13 (97) triumph at Marvel Stadium on Saturday.
Sydney (8-1) reclaimed top spot from Fremantle on percentage, while North (4-5) will end the round sitting below the wildcard positions.
The Swans were well served by Heeney (24 disposals, seven clearances), Chad Warner (18 disposals, one goal), Nick Blakey (26 touches) and ruck star Brodie Grundy, who was involved in an absorbing battle with Tristan Xerri.
Blakey provided his trademark run off half-back despite appearing troubled by a right shoulder issue.
Hayden McLean and Logan McDonald kicked three goals each in the absence of fellow tall forward Charlie Curnow (abdomen) and Tom Papley provided a spark with two.
Luke Davies-Uniacke (34 disposals), Luke Parker (29) and Harry Sheezel (31) fought hard for North, who had more scoring shots and more inside-50s (67-51) than their opponents, but couldn't make it count.
Finn O'Sullivan, Jack Darling and Zane Duursma kicked two goals each for the Kangaroos.
As North celebrated the 30-year anniversary of their 1996 premiership, with Wayne Carey among the former stars in attendance, Alastair Clarkson's current team stunned the Swans early.
Davies-Uniacke and O'Sullivan were prominent as the Kangaroos dominated stoppages and territory in a six-goal first term, leading by as much as 19 points.
But two late Swans goals trimmed the margin back to seven by quarter-time and the visitors added the first three majors of a see-sawing second term.
The Kangaroos led by one point at halftime, when they held a 36-24 advantage in inside-50s, before Papley and Warner ignited Sydney with clever snaps on goal.
They came as the Swans added 6.3 to 4.1 in the third quarter, but North refused to lie down and weren't out of the contest until after Cam Zurhaar narrowly missed a flying shot at goal in the final minute.
It was Sydney's 11th consecutive win over North, continuing a run of dominance that started in 2018.
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