Leading by 39 points in the third term, Essendon had to deal with a dramatic last-quarter surge from the Swans at Marvel Stadium, before holding on to win 11.5 (71) to 8.15 (63).
But the Bombers, who have defied dire pre-season predictions to move to 5-3, will have to deal with a fourth season-ending injury.
Defender Hayes grabbed his knee in agony after landing awkwardly in a marking contest with Sydney star Isaac Heeney late in the third quarter.
"Lewis Hayes looks like an AFL player, but unfortunately ... we'll wait to absolutely confirm, but it's almost certain that he's done an ACL," Essendon coach Brad Scott said.
"I don't know how you sort of handle all of this, other than to say when you face adversity, you've got to come through it
"It's just devastating for 'Hayesy' because he's worked so hard to get this opportunity, and he took it. I thought he looked right at home."
If the fears prove founded, Hayes will be Essendon's third ACL victim after ruck Nick Bryan and forward Tom Edwards, while Sam Draper (achilles) is also out for the season
The Bombers also face a nervous wait for their superstar captain Zach Merrett, who struck Swans defender Nick Blakey in the final quarter.
One of the genuine contenders for the Brownlow Medal, Merrett is almost certain to face scrutiny from the match review officer after Blakey underwent a head injury assessment.
Sydney didn't say post-game whether Blakey had passed his concussion test.
"I thought I tapped all ball, I was about to kick the goal," Merrett told Fox Footy.
"I haven't seen it, but I don't think there's anything to worry about."
Scott refused to comment when asked about the incident, while Swans coach Dean Cox replied: "Both people just going at the ball."
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.
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, as well as the dynamic Tom Papley, inside 50.
Cox refused to blame a lengthy injury list for Sydney's struggles.
Minor premiers and beaten grand finalists last season, Sydney have slumped to 3-6 after Cox took over as coach from John Longmire in November.
"Probably not as much about personnel, more about method," Cox said.
"(It's about) making sure that we engage their defenders as much as we possibly can.
"The rule we have is if you can't mark it, you don't get outmarked."
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