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.
Former Titans kicker dies in car crash
UPI Sports Results for September 20, 2014