Eberle knocked in a second-chance shot at 1:48 of the extra period after Justin Falk was whistled for boarding 19 seconds into OT.
The win gave Edmonton a 2-1-1 mark for their five-game homestand. Carolina lost for the second straight game.
Ryan Nugent-Hopkins scored in regulation for the Oilers as did Taylor Hall, Nail Yakupov and Jesse Joensuu, Devan Dubnyk recorded 24 saves.
Cam Ward allowed five goals on 29 shots for the Hurricanes.
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