After 70 minutes at Wembley dominated by Spurs, Pedro Obiang thundered a strike in off Hugo Lloris's near post from about 30 metres on the left with West Ham's first shot of the match.
Spurs went back on the offensive but their shots were blocked, saved or off target.
The one shot of the 31 they attempted that paid off came with six minutes remaining, as Son let fly from a distance similar to Obiang's goal to equalise.
Spurs pressed for a winner but West Ham held firm to go two points above the relegation zone in 15th.
The draw leaves Spurs fifth, three points off Liverpool in fourth.
Spurs: 13 Jan: v Everton (H)
West Ham: 13 Jan v Huddersfield (A)