With West Brom struggling for survival in the Premier League, Tottenham were expected to win the match quite handsomely.
Spurs carried the greater threat and Ben Foster needed to be alert when Erik Lamela threatened before he palmed away Christian Eriksen's free kick after 62 minutes.
Kane nearly became the Baggies' hero when a mis-hit clearance almost flew into the back of his own net with Hugo Lloris acrobatically keeping it out. Of course, disappointed after this result, but we can still achieve to be in the top four. That was thanks to a scruffy goal in stoppage time, with Jake Livermore claiming the final touch as he and defender Danny Rose stretched to reach the ball in a frantic scramble.
Chelsea can close the gap to two points with a home win over Champions League finalists Liverpool on Sunday, and they face Huddersfield Town and Newcastle United next week, with Spurs hosting Newcastle and Leicester City.
Substitute Nathan Redmond's first goal in nearly a year appeared to put the fight very much back in Saints' hands as it would have left them needing a win at fellow struggler Swansea on Tuesday to guarantee safety.
Then the Baggies enjoyed yet another incredible reprieve when a Southampton victory at Everton in the late game, which would have confirmed their relegation, was prevented by a 96th minute equaliser for the home side in a 1-1 draw.
"That was as loud a crowd as we've probably had this season and they were willing us on to the final whistle".
Stoke took the lead through Xherdan Shaqiri's 43rd-minute free kick, but James McArthur equalized for Palace in the 68th and Patrick van Aanholt netted the victor in the 86th.
Yet though referee Jon Moss had added on only four minutes, there were just over 95 minutes on the clock when Tom Davies' strike deflected in off Wesley Hoedt.
Palace lost its first seven games but is mathematically safe, completing the considerable upturn in its fortunes after Roy Hodgson succeeded Frank De Boer early in the campaign.
Less than three hours later, it seemed their heroics could be all in vain as Nathan Redmond's header for Southampton at Goodison put the Saints, who later had Maya Yoshida sent off, in sight of the three points that would condemn Albion.