Joel Matip's goal was his first in the Premier League this season
The Hammers took a 12th-minute lead when Lucas Paqueta, after playing a one-two with Michail Antonio, fired from 20 yards out past Liverpool goalkeeper Alisson.
But the advantage lasted less than six minutes before Cody Gakpo equalised with his own shot from outside the penalty area.
West Ham thought they had gone back in front with Jarrod Bowen's angled strike but the England forward was ruled offside by the video assistant referee to keep it at 1-1.
An unmarked Joel Matip scored what proved to be the winner in the 67th minute with a powerful header from Andy Robertson's corner from the left.
There was a late scare for Liverpool when Thiago handled in the penalty area, with Hammers boss David Moyes furious that a decision was not awarded by either on-field referee Chris Kavanagh or Neil Swarbrick on video assistant referee duty.
At the final whistle Moyes confronted Kavanagh, still angry at the decision that cost his side an opportunity to earn a point.
It has been a tough season for Liverpool, who came second in 2021-22, losing out to Manchester City by one point as the title race went to the final day of the season.
They went out of both the FA Cup and Carabao Cup, competitions they won last season, at the fourth-round phase and were hammered 6-2 on aggregate by Real Madrid in the last 16 of the Champions League.
Therefore, to miss out on European football altogether would be a massive setback for Jurgen Klopp's side.
They may have to settle for the Europa League at best and will need to finish fifth or sixth to qualify for the second-tier European competition, while seventh would leave them in the play-offs for the third-tier Europa Conference League.
This win takes them sixth and above Tottenham on goal difference, although Champions League football looks beyond the Reds as they are six points behind third-placed Newcastle and Manchester United in fourth, with both sides having at least one game in hand.
"If we win all our games, we will be in Europe," said Liverpool manager Jurgen Klopp. "People ask me about the Champions League but that's not in our hands as we don't play any of the teams above us.
"I would like to finish the season as us, then we can build on that next season.
"I want to take something out of this season. If that's European competition, that's great; if not, we have to accept it.
"For a few weeks, I want us to show our real faces instead of the nice one in one game, then the ugly one in the next."
After West Ham's 4-0 thrashing of Bournemouth at the weekend, Antonio said they needed one victory to be assured of staying in the Premier League.
The Hammers made a bright start at London Stadium as Paqueta got the game's first goal.
But the Reds, with Trent Alexander-Arnold excelling in a dual role - at right-back but with the freedom to move into midfield when Liverpool were in possession - levelled soon afterwards with his pass finished by Gakpo.
Lucas Paqueta has scored in each of his past two Premier League appearances for West Ham, having scored twice in his first 21 league games for the club
Diogo Jota could have made it 2-1 and should have done better when the ball rebounded into his path but shot over the top, and he also glanced a header just wide from four yards out.
However, West Ham had further first-half chances too. Liverpool's Virgil van Dijk denied Antonio a certain goal from close range with a crucial interception following Said Benrahma's angled ball.
The resulting corner did fall to Antonio but the Jamaica striker could only glance his header off target.
In the second half, Matip, who started in place of the injured Ibrahima Konate, had an effort blocked on the line by Lukasz Fabianski but then scored shortly afterwards with a powerful header when unmarked from a corner.
West Ham had opportunities to score an equaliser, but Tomas Soucek headed over and Bowen had a low strike saved as Liverpool held on for their third win in a row.
-- Courtesy of BBC Sport