Mauricio Pochettino's side, fresh from beating European champions Real Madrid in midweek, found Palace a difficult nut to crack at Wembley and could have been behind before Son curled in the winner midway through the second-half.
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Tottenham remained in third position with 23 points, level with Manchester United who play at fourth-placed Chelsea later.
Palace have four points from 11 games but could count themselves unfortunate as Scott Dann was twice denied by Tottenham's debutant keeper Paulo Gazzaniga while Wilfried Zaha wasted a great chance shortly before Spurs scored.
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Gazzaniga, playing in place of the injured Hugo Lloris, also saved well from former Tottenham player Andros Townsend early in the second half while Son could have given a nervy Tottenham breathing space but dragged a good chance wide.
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