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The Swans had most of the play in the thrilling final term, but spurned the opportunity to secure the points with an inaccurate 2.9 for the quarter that allowed the Bombers to hang on for the 15.10 (100) to 14.15 (99) win.
The Swans led by 12 points 20 minutes into the last quarter, but goals to Alwyn Davey and David Hille then a rushed behind and a point to Dyson Heppell set up the thrilling final minute that ended with Goodes marking the ball just inside the 50m arc.
However, the co-captain could not cap an excellent individual performance with the game-winning goal - dragging his kick to the left of the goal post.
It was a stunning end to an entertaining contest.
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The Swans' intent to exert maximum pressure on the opposition was evident from the outset with their superior work rate seeing them to an 18-point lead before Kyle Reimers booted his team's only goal of the term.
The Bombers worked into the contest in the second quarter and hit the lead for the first time at the 14-minute mark, as the Swans' intensity waned, but Goodes sparked a late rally that saw his team take a five-point lead into the main break.
The third term was an excellent example of the evenness of the teams with neither able to snatch a decisive break.
Essendon held a seven-point advantage going into the last quarter, but the Swans soon got their noses back in front despite Ryan O'Keefe's goal being sandwiched between six behinds.
Lewis Jetta appeared to have put the Swans back on track for a win with his long-range goal, but it would prove to be the set up for the stunning finale.
Essendon 1.2 8.5 13.7 15.10 (100)
Sydney Swans 3.3 9.4 12.6 14.15 (99)
GOALS
Essendon:Reimers 4, Hille 3, Zaharakis 2, Jetta 2, Melksham, Hurley, Stanton, Davey
Sydney Swans:Reid 3, Goodes 3, Jetta 2, Bird, Kennedy, Dennis-Lane, Mumford, McGlynn, O'Keefe
BEST
Essendon: Watson, Fletcher, Ryder, Heppell, Reimers, Hille, Hocking
INJURIES
Essendon:Nil
SUBSTITUTES
Essendon:Jake Melksham replaced by Michael Ross in the fourth quarter