The Swans' gun forward collected a bouncing ball from a pack and threw it on his boot from just outside the goalsquare with 30 seconds remaining to level the scores at 54 apiece at the SCG on Friday night.
But the scoreline might have been very different for the hosts who paid the price for inaccurate goalkicking throughout to finish with 6.18 (54) to the Cats' 7.12 (54).
Robbie Fox looked set to sum up Sydney's night when he ran into an open goal when trailing by two points at the 23-minute mark of the final term but slammed the ball into a post.
Geelong were on track to hang on for an unlikely victory after being outplayed for much of the contest until Heeney's final intervention meant last year's grand finalists shared the four points.
The Swans spread around their goalkicking yips, though Fox arguably spared Tom Hickey's blushes after the ruckman missed two of the easier set shots of the night from close range.
The troubles near goal started early for Sydney as they dominated the ball and territory in the opening term to lead 109-65 in disposals and inside 50s (17-9), but were wasteful with their 10 shots.
The Swans went into the first change trailing by nine points and the second term played out much the same way, as Sydney had another 11 shots on goal and took a 4.12 to 4.4 lead into the main break.
The game was more of an arm wrestle from there as the Cats slowed the Swans' ball movement and tightened up in defence.
Zach Tuohy starred for Geelong as he celebrated equalling Jim Stynes' record for the most VFL/AFL matches from an Irishman with 28 disposals.
Gryan Miers was also prolific with 26 disposals, while Mitch Duncan (25) looked back to his best in a third match since overcoming a hamstring injury.
Spearhead Tom Hawkins and sharpshooter Oliver Henry both booted two critical goals for the Cats, while Logan McDonald and Braeden Campbell finished with two for Sydney in a low-scoring contest.
The Swans next face Richmond in what looms as a must-win clash for both teams at the MCG on Thursday night, while Geelong host North Melbourne on Sunday.
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