Keanan Bennetts gave Spurs a third-minute lead but Domingos Quina equalised in the 20th minute.
Shilow Tracey quickly restored the visitors' lead and a valuable win looked on the cards.
But the Hammers hit back and earned a point when Marcus Browne, pictured above right, converted a 77th-minute penalty.
Spurs are 10th on goal difference ahead of Sunderland, who drop into the bottom two. Both sides are four points ahead of Manchester United.
West Ham remain sixth.
West Ham: 17 Feb v Liverpool (A) PL2
Spurs: 16 Feb v Swansea (A) PL2