Weber tallied 1:29 into the extra session to hand the Devils their 12th overtime loss of the season, most in the Eastern Conference.
David Legwand forced overtime by putting a slap shot past Cory Schneider with only 11 seconds to play in the third period.
Weber gave the Predators the early lead 6:45 into the contest.
Carter Hutton collected the victory with 24 saves. Schneider gave up the three goals on 33 shots.
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