Less than a minute into the extra frame, Colin Wilson unleashed a shot that missed the cage, but it rebounded off the backboards directly to an unmarked Weber, who fired it into an empty net for the decisive tally.
Weber also recorded an assist while Craig Smith, Gabriel Bourque and Roman Josi found the net for the Predators, who snapped a two-game losing streak.
Pekka Rinne stopped 26 shots in the victory.
Jimmy Howard started in goal for Detroit but left with an injury in the first period after yielding two goals on just seven shots.
Jonas Gustavsson relieved and stopped 16-of-18 tries.