Craig Dawson's 32nd-minute own goal put Tottenham in front as they sought to cut Leicester's lead back to five points.
But Tottenham produced a nervy second-half display and Dawson redeemed himself when he headed his side level after 72 minutes to deflate the home crowd and no doubt kickstart a party in the east Midlands.
Tottenham struck the woodwork three times on a frustrating night, and Leicester can now clinch the title if they beat Manchester United at Old Trafford on Sunday.
With three matches remaining Leicester have 76 points, while Tottenham, the only side who can deny them the title, are on 69.
Manchester City and Arsenal have 64.
"It was very frustrating," Tottenham manager Mauricio Pochettino told reporters. "It was a game we had to win.
"The first half we played very well but we spent a lot of energy to make it 1-0 and in the second half we lost a bit of control of the game and that allowed them to believe."
Asked if Tottenham had blown their title chances, the Argentine added: "It's not over, the gap is seven points and we have three games left. But it's true it's very difficult."
Tottenham bossed the opening period and West Brom barely got out of their own half in the opening 15 minutes.
Leading scorer Harry Kane saw his close-range shot turned onto the inside of the post by West Brom goalkeeper Boaz Myhill and then Tottenham's Danish playmaker Christian Eriksen grazed the crossbar with a free kick.
West Brom were well organised, however, and had managed to blunt Tottenham's attack until the opener arrived in unusual circumstances.
Eriksen's dangerous free kick was aimed towards Jan Vertonghen but Dawson, in his desperation to intercept it, stumbled and saw the ball go into the net off his arm.
There was relief all around a chilly White Hart Lane as the home fans again began to dream of a first title since 1961.
But the second half did not go to plan.
West Brom were far more ambitious, causing Tottenham problems with a barrage of high balls into the area.
Even so Tottenham still should have doubled their lead when Eriksen and Dele Alli combined to tee up Erik Lamela but his close-range shot spun against the foot of the post.
Eriksen curled another effort just wide but the second goal would not arrive and the nerves began to jangle.
Salomon Rondon headed wide when unmarked and had another shot tipped over by Hugo Lloris. From the resulting corner the ball was whipped in and Dawson rose above Eric Dier to head past a flat-footed Lloris.
Tottenham were stunned and never really looked like snatching a late winner that would have kept the pressure firmly on Leicester. While Pochettino refused to admit defeat, his side's priority now will be to consolidate second place.
