Gareth Bale was close to giving Spurs the lead after only eight minutes but his volley sailed over the Baggies goal.
Rafael Van der Vaart was closest to opening the scoring in the first period but he failed to hit the target after some good work by Aaron Lennon on the right wing.
Ben Foster was alert to keep the hosts frustrated before the break as he reacted well to deny Jermain Defoe from close range.
West Brom started to assert themselves in the second period and Brad Friedel was called into action to block substitute Romelu Lukaku's powerful effort.
Marc Antoine-Fortune was gifted an opportunity by some poor defending but Friedel dived to his right to prevent the striker from putting the visitors ahead.
Spurs were surprised by the Baggies in the second period but Bale almost stole a lead with a dipping strike from 30 yards that was beaten away by Foster.
The introduction of Emmanuel Adebayor seemed to encourage the hosts and they were ahead after 74 minutes when Benoit Assou-Ekotto's drive was deflected before flying past Foster to give Spurs the lead.
Andre Villas-Boas's team looked comfortable after making the breakthrough but were denied victory in the final minute when James Morrison pounced inside the area to fire in an equaliser.