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Dele Alli's equaliser a minute from time ensured Tottenham maintained their unbeaten start to the season and remained within a point of Premier League leaders Manchester City.
Nacer Chadli's late goal for West Brom seemed set to inflict an unlikely first defeat of the season on in-form Spurs.
But Mauricio Pochettino's men fought back as substitute Son Heung-Min and Christian Eriksen combined to create the opportunity for England international Alli to prod the ball home with the outside of his boot.
It was thanks to a series of fine first-half saves from Ben Foster, 33, that the hosts were still in contention at the break, with Alli admitting his team could have scored "three or four" were it not for the former England goalkeeper.
But despite enjoying less than 30% of possession, West Brom were worthy of a draw, improving considerably as an attacking force after the interval.
Eriksen could have snatched victory with a free-kick aimed towards the top-right corner in the last action of the match but, once again, the outstretched Foster kept out the Dane's curling effort.
The draw ensures Tottenham are unbeaten in the opening eight league matches of the season for the first time since 1990, but it also allowed north London rivals Arsenal to leapfrog them into second in the table with victory over Swansea.
It was widely thought Tottenham gave their best display of the Mauricio Pochettino era in beating Manchester City 2-0 on their last league outing.
This time they lacked the verve they displayed in that win over City, but the determination they displayed confirms their title credentials in what is their best start to a top-flight season since they began their 1960-61 title-winning campaign with 11 straight wins.
"We're still unbeaten and that's the important thing," said their Argentine manager, who praised West Brom's Foster for a "great performance".
Tottenham dominated the first half - enjoying 76% of possession and creating 10 shots on goal (five of which were on target).
The excellent Foster dived low to prevent Ben Davies from finding the bottom-right corner, twice denied Eriksen and also stopped Alli from putting Spurs ahead when the England forward had been allowed to surge forward unchallenged.
Worrying perhaps for the visitors, however, was the second-half injury to Toby Alderweireld which forced the influential defender to leave the pitch on a stretcher.
It was an innocuous collision with his team-mate Jan Vertonghen, the latter clipping his fellow centre-back's knee, which ended the Belgian's participation.
Predictably, West Brom sat deep and frustrated Spurs. Starved of possession - they made 104 passes in the first half compared to Tottenham's 345 - Tony Pulis' men had to rely on a Tottenham mistake for their best chance of the opening period.
When Hugo Lloris failed to safely gather James McClean's long-range shot, Salomon Rondon had a chance to take advantage three yards from the goalline, but the France goalkeeper saved himself by swiping the ball away from the home striker's feet.
Foster admitted after the final whistle his team never had any intention of vying with their opponents for possession.
"It's a classic Tony Pulis team," said the goalkeeper. "When you come up against teams like Tottenham, you know you won't be able to outplay them."
But the hosts weathered the first-half storm and posed more problems after the break with Gareth McAuley squandering a fine chance in front of goal before Chadli struck for his fourth goal in five Premier League games.
With less than 10 minutes remaining, Chadli pounced on a McClean strike which rebounded off the bottom of the post. It proved not to be the winner, but the goal did help extend West Brom's unbeaten league run to four matches.
Tottenham boss Mauricio Pochettino: "I'm a little bit disappointed because we dominated the game and created better chances, but we know every time we play West Brom it's going to be difficult.
"In the first half we dominated but the same thing always happens in football if you don't score. Foster was man of the match. It was difficult when we conceded the goal and the team showed character."
