At the bottom of the table, Norwich were relegated as Michail Antonio's four-goal blitz fired West Ham to a thumping 4-0 win that should seal the Hammers' top-flight status for another season.
Sheffield United have long since achieved that feat in their first season in the Premier League for 13 years and the Blades are still eying the top four themselves as they moved within five points of fourth-placed Leicester.
Chelsea remain in third, but could end the weekend down in fifth if Leicester and Manchester United beat Bournemouth and Southampton respectively.
The Blues have already reinforced their attack for next season with the signings of Timo Werner and Hakim Ziyech, but Frank Lampard was again shown it is at the other end his side desperately need to improve.
David McGoldrick's long wait for a Premier League goal came to an end when he was quickest to react after Kepa Arrizabalaga saved Oli McBurnie's deflected shot.
More slack defending from the visitors then allowed McBurnie a free header to double the hosts' lead.
Lampard sent on Antonio Rudiger at the start of the second-half, but the German international did little to improve Chelsea's defending as his attempted clearance from Lys Mousset's cross only teed up McGoldrick to score his second 13 minutes from time.
Chelsea will expect to get back to winning ways when Norwich visit Stamford Bridge on Tuesday, but then face Liverpool and Wolves in a tough final week of the season to leave their prospects of finishing in the top four in the balance.
- Liverpool held -
Liverpool will still be motivated to beat the Blues despite already clinching the title as Jurgen Klopp's men now need to win their remaining three games to be surpass the record total of 100 points in a Premier League season set by Manchester City two years ago.
For the first time since January 2018, the Reds failed to win a Premier League game at Anfield as an inspired performance by Nick Pope in the Burnley goal and Jay Rodriguez's equaliser to cancel out Andy Robertson's first half header gave the Clarets a share of the spoils.
"For moments it was Liverpool against Pope. We did everything right and he made saves but we should have scored more, that's on us," Klopp said.
"We didn't close the game and they took their moment. We were angry with the referee but we have to criticise ourselves first for not finishing the game."
At Carrow Road, a seventh successive league defeat condemned Norwich to relegation as Antonio's ruthless display boosted West Ham's own survival bid.
Daniel Farke's side are 13 points from safety with only three games left, meaning they are certain to make an immediate return to the Championship after last season's promotion campaign.
"We wanted to beat the odds again but when the dust settles the outcome is more or less what was expected. It doesn't take anything away from the fact that we are disappointed," Farke said.
"When we are 100 percent we are competitive but when it's 96 or 97 percent then it sometimes looks like men against boys. That's what I expected."
West Ham's second win in four games opens up a six-point cushion over the bottom three.
The early signs were ominous for Norwich as West Ham took the lead in the 11th minute through Antonio's close-range volley.
Antonio struck again in first-half stoppage-time, glancing his header into the far corner from Mark Noble's free-kick.
He completed his hat-trick in the 54th minute with a looping header and his fourth came in the 74th minute from Ryan Fredericks' cross.
Watford also moved to the brink of survival as two Troy Deeney penalties beat Newcastle 2-1 to move level with West Ham on 34 points.