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.
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