It could have been all three points for the Baggies after Zoltan Gera fired them into a first-half lead, but Robbie Keane snatched a second-half equaliser to also keep alive their hopes of achieving a top-seven finish.
Head-coach Martin Jol was looking for a fourth successive home victory in the Premiership as his side were strongly challenging for a UEFA Cup qualifying spot.
Jol was forced to make one change from the side which earned a credible 2-2 draw against Liverpool last weekend, with Jermain Defoe replacing the injured Frederic Kanoute who was ruled out with a stiff neck.
Albion manager Bryan Robson was also hoping his side could maintain their impressive recent run which has seen them suffer just one defeat in their last six games and given them a strong chance of avoiding relegation.
Despite languishing at the foot of the table at Christmas, Robson has worked his magic at The Hawthorns and could yet create Premiership history if they achieve this amazing act of escapology.
The former Manchester United captain stated before the game he felt his side needed three more wins if they were to achieve the unthinkable.
And they almost made a dream start at White Hart Lane when Kanu found himself unmarked inside the box after just three minutes. The Nigerian international saw his close-range shot blocked by Erik Edman.
The rebound fell to Hungarian midfielder Gera who drilled his shot wide of the post.
There seems to be a new sense of belief in the West Brom side and they showed it once again on ten minutes when Neil Clement's superb 25-yard free-kick struck the outside of the post.
However, from the resulting goal-kick Tottenham's Michael Carrick planted a deep cross into the box which was almost turned into his own goal by Thomas Gaardsoe. Russell Hoult made a fantastic reflex save to finger-tip the ball away.
But West Brom were rewarded for their early pressure when they opened the scoring on 23 minutes. Richard Chaplow's corner was headed away by Michael Dawson, only into the path of Gera. The Hungarian midfielder took two touches before striking a delicate left-shot past Paul Robinson into the corner of the net.
And West Brom should have made it two on 30 minutes. Kanu raced away down the right before crossing the ball in to Chaplow, whose shot was superbly beaten away by Robinson.
Stephen Kelly came close to snatching an equaliser for the home side on 35 minutes when Andy Reid's corner picked out the Irish defender, but his header flew just over the crossbar.
Jol was clearly not happy with his side's first-half display and signalled his intentions at half-time when he made a double substitution.
Thimothee Atouba and Mido were brought on to replace Edman and the ineffective Sean Davis.
Mido almost made an immediate impression just two minutes into the second half when he turned Ronnie Wallwork and Gaardsoe inside the box, but drilled his shot into the sidenetting.
Jol's change of tactics and the aerial dominance of Mido worked to his advantage on 52 minutes when Keane scored the equaliser.
Atouba's high cross was only half-headed away by Clement who was strongly challenged by Mido. The ball fell straight into the path of Keane who made no mistake striking a sweet shot past the stranded Hoult for his 17th goal of the season.
Robson could not believe his luck and must have thought it was not going to be his day when straight from the kick-off, Junichi Inamoto was released clean through into the box, but saw his low drive saved by Robinson diving low down to his right.