Manchester United's push for Europe was dented by Burnley, who secured a 1-1 draw at Old Trafford, keeping alive their hopes of a dramatic escape act.
Klopp's glittering reign at Anfield appears to be ending on a flat note after they dropped points for the fourth time in five league games at the London Stadium.
Just weeks ago Liverpool, who won the League Cup in February, were on course for a quadruple of trophies but that is now a distant memory after FA Cup and Europa League exits and a string of damaging league results.
Liverpool would have moved level on points with leaders Arsenal with a victory, at least temporarily, but they failed to make the most of their chances despite a stirring second-half fightback.
The Gunners are on 77 points, two clear of Liverpool with a game in hand, while defending champions Manchester City have 76 points with two games in hand on the Reds.
Jarrod Bowen put David Moyes's West Ham ahead shortly before half time but Liverpool looked a different side in the second period, taking the lead through Andy Robertson before an own goal from West Ham goalkeeper Alphonse Areola.
But West Ham levelled in the 77th minute when Bowen wriggled free down the right and lifted a cross towards Michail Antonio, who headed home to make it 2-2.
A downbeat Klopp, in the final weeks of his time at Anfield, refused to talk about his side's title chances.
"I'm not in the mood to talk about that to be honest or look at that at all," he told TNT Sports.
"We had to win here, we knew that, and we didn't, and now we have a little bit more time between now and the next game. We try to get the boys ready again and we will go again."
Klopp attempted to play down a touchline spat with Mohamed Salah, who was left out of the starting line-up, but the Egypt forward fanned the flames.
"There's going to be a fire today if I speak," he told reporters.
Pep Guardiola's City, who play relegation-threatened Nottingham Forest on Sunday, are hot favourites to secure an unprecedented fourth straight English top-flight title.
Arsenal, who led the way for much of last season, are firmly in the hunt but face a tough away day at Tottenham, also on Sunday.
Sheffield Utd down
Newcastle's 5-1 win against basement club Sheffield United strengthened their push for European football next season but sent Chris Wilder's team down to the Championship after just one season in the top flight.
Anel Ahmedhodzic gave the visitors an early lead at St James' Park but Alexander Isak scored twice as the home side roared back.
"We have not been good enough," Wilder told the BBC. "The league has been too powerful for us right the way through the season."
Newcastle remain in seventh place but are just one point behind Manchester United, who conceded a late penalty in a draw against Burnley.
Brazilian winger Antony scored his first Premier League goal of the season but Zeki Amdouni levelled from the spot in the 87th minute to leave the home side ruing missed chances.
United have now won just one of their past six league matches as they limp towards the end of the season.
Burnley are two points behind Forest, who are just outside the relegation zone, and one point behind Luton.
Ten Hag said his team "gave it away".
"We are one of the most dynamic and entertaining teams in the league at this moment," said the Dutchman.
"We are creating loads of chances by playing good football. it was unnecessary to lose control."
Hwang Hee-chan and Toti Gomes scored as Wolves deepened Luton's woes with a 2-1 win and Fulham drew 1-1 with Crystal Palace.
Everton host Brentford later on Saturday seeking their fourth win in five games while Aston Villa, chasing a Champions League spot, take on inconsistent Chelsea in the late kick-off.
