A week later, they saw Stoke relegated with a 2-1 loss to Crystal Palace.
But the host dug in for long periods against Spurs and its persistent set-piece threat brought reward in the 92nd minute when England midfielder Livermore scrambled home from virtually on the goal line.
Defeat meant Tottenham have yet to confirm its top-four spot.
And with Spurs' fourth place looking suddenly vulnerable Pochettino insisted they need to be ruthless with games against Newcastle and Leicester left.
Stoke had hope of staying up, when Xherdan Shaqiri scored with a 43rd-minute free-kick, but James McArthur equalized in the 68th minute and Patrick van Aanholt hit the victor with four minutes left.
Tottenham's defeat left them fourth on 71 points and gave Chelsea new hope that they might steal the fourth Champions league spot if they can move on 69 points by beating Liverpool at Stamford Bridge on Sunday.
Palace on the other hand mathematically sealed their Premier League status for another season, becoming the first team to avoid the drop from the top-flight despite losing their opening seven league games since 1899/00 (Liverpool, in case you were wondering). The two teams immediately above it, Swansea and Southampton, have played a match less and meet each other on Tuesday.
Albion were running out of time but Kane nearly scored a spectacular own goal when Lloris turned his sliced clearance over with 17 minutes left.More news: Bye bye Delhi Daredevils! See you next year
Brighton secured English Premier League football for another season by beating Manchester United 1-0 on Friday.
Bournemouth broke the deadlock eight minutes from halftime.
It was Swansea's seventh game without a win, and third in a row without a goal. Of players with 15 goals or more, only David Ginola can match that.
Fellaini had the ball in the net after only three minutes but he was offside when he turned home Rashford's free kick.
In the first half of few clear-cut chances, Kane had the best of Spurs' opportunities and his header from Erik Lamela's cross was saved by Ben Foster.
That was about as good as it got in a horribly disjointed display from United.
Southampton broke the deadlock 11 minutes into the second half when Redmond directed a far-post header between Pickford's legs from Cedric Soares' cross, his first goal since May 13.
He is certainly one of the most entertaining - 67 per cent of his 21 Premier League goals have come from outside the penalty area.
When Newcastle finally produced a move of substance, after the break, they found the net.