Andre Villas-Boasâ€™ first home game in charge did not finish the way he would have liked after Spurs took the lead in the 74th minute through Benoit Assou-Ekotto.
The left-back hammered home a half-volley from close range thanks to a deflection off Baggies defender Gareth McAuley after Gylfi Sigurdssonâ€™s cross had fallen into his path.
Earlier, Spurs had the ball in the back of the net in the 64th minute when Jermain Defoe finished Assou-Ekottoâ€™s cross but the goal was disallowed as the linesman ruled the England striker off-side.
It was a frantic finish in the final few minutes and after Morrisonâ€™s leveller, Spurs new boy Jan Vertonghen thought he had the winner when he volleyed home but an earlier infringement by Gallas forced referee Mike Dean to call the ball back, denying Spurs maximum points.
White Hart Lane, London â€“ 36,166
Yellow cards: TOT - Jan Vertonghen 48â€™
WBA - Youssuf Mulumbu 30â€™, Steven Reid 81â€™