Moore engineered a 12 1/2-minute drive leading up to the game-winning, one-yard toss to Clay with 10:27 left in the game, which put the Dolphins ahead for good at 13-10.
The Dolphins (6-10) ended the season by winning six of nine games.
New York (8-8), meanwhile, failed in its bid for a club-record third straight year with a playoff appearance.
Sanchez was intercepted three times overall with a late touchdown pass in the setback.
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