David Legwand had levelled things up for the Predators with just 11 seconds left to play in the third after Patrik Elias and Jaromir Jagr scored in the second to put the Devils on top.
Weber, who opened the scoring in the first, scored 1:29 into OT to settle it for Nashville.
Goals from Brian Boyle, Derick Brassard, Brad Richards, and Ryan McDonagh gave the New York Ramgers their second win over the New York Islanders in as many games.
Kyle Okposo tied it at 1-1 for the Islanders, who also lost two nights previously at Yankee Stadium.
There were also wins for the Winnipeg Jets, the Detroit Red Wings and the Carolina Hurricanes.