The game was scoreless until 5 minutes were left in the final frame when Washington's Troy Brouwer put the Capitals ahead on a wrist shot from the high slot.
Weber added an empty-netter for Nashville, which snapped a five-game losing streak against Washington.
Pekka Rinne turned aside 39-of-40 shots to earn the victory.
Washington's Tomas Vokoun matched him until third period, stopping 28-of-30 Predators shots as the Caps lost for the fourth time in five games.
Boston schools pull out free condoms over wrapping complaints
Aaron Carter is still in love with Hilary Duff