Spurs dominated for much of the fixture in the West Midlands but in the second minute of added-on time, Livermore netted from close range to keep the Baggies' hopes of a miracle survival alive with a week remaining of the season.
As for Spurs, they find themselves looking over their shoulder to fifth-placed Chelsea, who will move to within two points of a top-four spot with victory over Liverpool on Sunday.
From kickoff, Spurs showed their intent to dominate possession and West Brom were prepared to give the visitors plenty of time on the ball while keeping 10 men in their own half.
The strategies of both teams resulted in an uneventful opening quarter to the game with Spurs not registering their first effort on goal until Victor Wanyama saw a long-range strike tipped over the crossbar by Ben Foster.
Shortly afterwards, Kieran Trippier threaded a ball through to Harry Kane down the right-hand side of the penalty area but despite a well-struck effort, Foster was able to divert the shot wide of the far post.
West Brom had been outplayed for much of the first half but the hosts should have taken a lead into the break, with Jay Rodriguez seeing a shot cleared off the line by Jan Vertonghen after getting the better of Wanyama in the air.
After the restart, West Brom continued on the front foot but it was Spurs who created the first chance of the second half with Kane heading harmlessly over the crossbar from a promising position.
Despite requiring a goal to remain in the division, West Brom continued to line up defensively but Moore's side almost got the good fortune that they were looking for when Kane nearly put through his own net from five yards out, with Lloris making a brilliant save to keep the ball out.
During the closing stages, Moore made changes which were not warmly received by the home supporters with Salomon Rondon and Rodriguez - the club's two top goalscorers - being withdrawn, and a third alteration followed with Grzegorz Krychowiak replacing James McClean.
However, in the second minute of added-on time, West Brom found an unlikely winner. A set piece was sent into the box and after a goalmouth scramble ensued, Livermore was on hand to prod home from close range to keep West Brom's hopes of the most unlikely survival alive for at least three more days.
WEST BROMWICH ALBION (4-5-1): Foster; Nyom, Dawson, Hegazi, Gibbs; Livermore, Brunt; Phillips, Rodriguez (Chadli 79'), McClean (Krychowiak 87'); Rondon (Sturridge 74')
TOTTENHAM HOTSPUR (3-4-3): Lloris; Sanchez, Alderweireld, Vertonghen (Llorente 85'); Trippier, Wanyama, Eriksen, Rose; Lamela (Lucas Moura 78'), Alli (Son 67'), Kane