Down by 39 points in the third quarter, Sydney suddenly found form late in that term to set up a thrilling finish at Marvel Stadium.
Essendon didn't score a goal after midway through the third quarter, but somehow prevailed 11.5 (71) to 8.15 (63).
But the Bombers will face a nervous wait for their superstar captain Zach Merrett, who struck Swans defender Nick Blakey in the final quarter.
One of the favourites for the Brownlow Medal, Merrett is almost certain to face scrutiny from the match review officer after Blakey underwent a concussion test.
"I thought I tapped all ball, I was about to kick the goal," Merrett told Fox Footy.
"I haven't seen itn but I don't think there's anything to worry about."
Sydney were their own worst enemies as inaccuracy at goal and poor ball use when going forward ultimately cost them the game.
The Swans won the inside-50 count 61-45, but Essendon's defence stood tall as they moved to 5-3 and inside the top eight.
Fit-again Zach Reid continued his meteoric rise this season, starring as Essendon's most important key defender.
Vice-captain Andrew McGrath, the dashing Mason Redman and Nic Martin also provided valuable run.
The result was soured after Bomber debutant Lewis Hayes went down with a suspected serious knee injury late in the third quarter.
Hayes grabbed his knee in agony after landing awkwardly in a marking contest with Sydney star Isaac Heeney.
Young Essendon forward Nate Caddy was the leading goal-kicker on the ground with three, while Merrett managed to kick two despite having a quieter game than normal in the midfield.
Will Hayward was Sydney's most threatening forward, even after suffering a heavy knock to his shoulder.
But the Swans are badly missing injured key forwards Joel Amartey and Logan McDonald inside 50.
Minor premiers and beaten grand finalists last season, Sydney have slumped to 3-6 after Dean Cox took over as coach from John Longmire in November.
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