Spurs dominated the first half but Palace goalkeeper Wayne Hennessey made three fine saves to keep the visitors in the game.
Palace midfielder Yohan Cabaye missed a good chance on the hour, blazing a shot wastefully over the bar after being teed up by Spurs old boy Andros Townsend.
Spurs striker Vincent Janssen also spurned an excellent opportunity when he shot wide after being sent clear by a fine pass from substitute Dele Alli but Wanyama headed in from close range following a corner to seal the points seven minutes from time.
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(Reporting by Ed Osmond; Editing by Tony Jimenez)
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