Gazzaniga, in for the injured Hugo Lloris and Michel Vorm, produced a superb save from Scott Dann's first-half header and after the break stopped efforts from Andros Townsend and Luka Milivojevic.
Wilfried Zaha had a good chance when he rounded Gazzaniga but could only shoot wide from a tricky angle before Spurs took the lead on 64 minutes when the ball fell to Son on the edge of the area and he fired home with his left foot.
Palace pushed for an equaliser but Spurs were able to hang on for their fifth win in six PL matches, while the Eagles remain at the bottom of the table, five points from safety.
Spurs: 18 Nov v Arsenal (A)
Crystal Palace: 18 Nov v Everton (H)