Former Tottenham midfielder Livermore struck deep into stoppage time to send the home fans at The Hawthorns into raptures after both sides had missed good chances in a scrappy game dominated by the visitors.
Now, having taken a seemingly doomed quest for Premier League survival into the final week of the season, the hugely popular Moore has become a serious contender to keep the job in which he has exceeded all expectations as a caretaker.
They avoided relegation again when Southampton conceded an injury-time goal to draw 1-1 at Everton in Saturday's late game.
West Brom have dropped 26 points from winning positions this season - the most in the league - and there were few hints they would take the lead at The Hawthorns.
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.
Tottenham now occupy the final spot for the European competition and would have seen a match against West Brom as an opportunity to build the gap between them and Chelsea, however, things did not go to plan.
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Several protesters were handed long prison terms after the rally, which led to a split in the opposition's anti-Putin coalition. Putin returned to the presidency in 2012 and was re-elected to a fourth term earlier this year.
Though Southampton were left fuming that the goal had been scored after the allotted four minutes of added time, the point still moved the Saints out of the drop zone on goal difference ahead of Swansea City, who lost 1-0 at Bournemouth.
The result leaves fifth-place Chelsea just five points behind Spurs with a game in hand, and a frustrated Pochettino called for more from his players in the final two matches of the campaign.
Defeat meant a significant setback to Tottenham's hopes of claiming a top-four finish.
A week later, they saw Stoke relegated with a 2-1 loss to Crystal Palace.
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.
Yet Roy Hodgson's resurgent side, dominant after the break as Stoke were gripped by nervy caution, equalised after a sharp counter-attack through James McArthur before a Ryan Shawcross mistake let in Patrick van Aanholt for the 86th-minute victor. That game ended 1-1.
The win moved David Moyes' team six points clear of the bottom three.