Livermore prodded the ball home in the 92nd minute at The Hawthorns to leave Mauricio Pochettino’s men trailing Liverpool in the race to qualify automatically for next season’s Champions League group stages.
In fact, an unlikely defeat kept open the possibility of Spurs missing out on the top four in the Premier League, with Chelsea in with a hope of catching their London rivals.
But for West Brom, victory meant they could yet beat the drop as they made it five games unbeaten, including three wins, under caretaker manager Darren Moore.
They still, however, need a fortunate combination of results elsewhere to take their fight into the final day at Crystal Palace.
A sluggish opening 20 minutes created a clear impression that neither side had the intensity of clubs with lots to play for.
There was more focus on off-field events in the opening stages, with Tottenham fans singing their support for Toby Alderweireld and Pochettino.
Defender Alderweireld was making just his second appearance since February as his contract dispute with the club rumbles on while cryptic comments from Pochettino in recent weeks have called his own future into question.
– Defiance –
Meanwhile, the sound of an aeroplane chartered by Wolves fans circling overhead to poke fun at Midlands rivals West Brom with a banner reading ‘Boing Boing Bye Bye’ was the cue for a show of defiance from home supporters.
The first serious chance arrived in the 24th minute when Spurs moved the ball to the edge of the Albion area, from where Victor Wanyama struck a bending shot that drew a smart save from home goalkeeper Ben Foster.
After Danny Rose had fired wide from the resulting corner, Harry Kane had two chances to open the scoring for Tottenham.
First he collected a neat pass from Kieran Trippier but was thwarted by Foster, although the keeper knew little about the save as the ball cannoned off his forehead.
Then Kane met a Christian Eriksen cross from the left but headed wide under pressure from Ahmed Hegazi.
Yet just before half-time Spurs were forced to deal with some unexpected pressure, first when Alderweireld was forced to head a Jay Rodriguez cross over the bar and then when Hegazi headed over from a Matt Phillips corner.
West Brom should have led early in the second half when Phillips burst down the right and crossed for Salomon Rondon, but his low shot was blocked by Davinson Sanchez.
But it was Spurs’ turn next as they produced a sweeping move that presented a chance to Erik Lamela.
He collected a neat flick from Dele Alli but was blocked bravely by Foster as he bore down on goal.
Both Tottenham’s Danny Rose and Albion’s Allan Nyom received yellow cards for an incident that reflected badly on both players.
Rose was fortunate to escape sterner punishment for putting a hand in Nyom’s face, but the Albion defender’s over-the-top reaction also left a sour taste.
With results elsewhere going their way, West Brom began to scent the goal that would give them hope of taking their survival fight into the final week of the season.
Hugo Lloris in the Tottenham goal had to make a fine save to spare Kane the embarrassment of an own-goal before Alderweireld made an important clearance from a Hegazi cross-shot.
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