Left smarting after falling short of a last-gasp victory over Essendon, the Swans dug deep to overcome a slow start and secure an 11.12 (78) to 9.8 (62) win at the SCG on Friday night.
The result keeps Sydney in touch with the top eight, sitting 11th (4-6) ahead of the weekend's matches.
Heeney was awarded the Goodes-O'Loughlin medal as best on ground in the annual Marn Grook match, finishing with a monster haul of two goals, 38 disposals and 10 clearances.
Meanwhile, Carlton remain 10th after suffering their fourth-straight SCG defeat since 2021.
Adam Cerra led the Blues with 33 disposals, while Charlie Curnow slotted three majors to reach then surpass the 300-goal milestone.
The visitors started the cleaner team with 20 inside-50s to Sydney's nine in the first quarter, but could only lead by six points at quarter-time.
Patrick Cripps, after kicking two goals in the opening term, sent a scare through the Blues camp when he limped off in the second quarter.
The two-time Brownlow medallist looked to have injured his left ankle during a marking contest with Chad Warner, but was able to return after receiving treatment on the bench.
Heeney and Justin McInerney slotted a goal each to level the scores at 5.3 (33), before Harry McKay (two goals) hit back to restore Carlton's lead before the main break.
After struggling to break away in a third-quarter slog, the Swans found their spark when a streaming Brodie Grundy launched the home side ahead before Joel Amartey added another.
Amartey, returning from a hamstring injury, could come under scrutiny from the match review officer for his fourth-quarter bump against Carlton defender Jordan Boyd.
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