Sheffield United dealt a major blow to Chelsea's chances of a Premier League top-four finish by thrashing the Blues 3-0 at Bramall Lane, while Liverpool's long winning run at Anfield came to an end in a draw with Burnley on Saturday.
As it happened | Sheffield United v Chelsea
Manchester City did guarantee their presence in the top four for a 10th successive season with Raheem Sterling scoring a hat-trick in a 5-0 win at Brighton.
However, they will have to wait for the outcome on Monday of their appeal against a two-season ban from European competitions to see if they will play Champions League football next season.
Norwich will be playing in the Championship after Michail Antonio's four-goal blitz confirmed the Canaries' relegation in a thumping 4-0 win for West Ham that should seal the Hammers' top-flight status for another year.
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 first 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.
"They were better than us, physically, in the mind and with the ball," said Lampard. "I learned a lot and I won't forget that."
For the first time since January 2018, Liverpool 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 Burnley a share of the spoils.
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 City two years ago.
"We left the door open, we didn't close it. We should have scored two, three, four goals at least," said Klopp.
City scored four or more goals for the fourth time in six league games at Brighton.
Sterling's treble took his best ever season's tally to 27, while Gabriel Jesus and Bernardo Silva also netted in another rout for a pleased Pep Guardiola, who is hoping his players' efforts on the pitch are rewarded by playing Champions League football next season.
"We deserve to be there because we won it on the pitch. Hopefully on Monday UEFA can allow us to play like this team and these players deserve to," said Guardiola.
At Carrow Road, a seventh successive league defeat condemned Norwich to relegation as Antonio's ruthless display boosted West Ham's own survival bid.
"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," manager Daniel Farke said.
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.
Abraham should have given Chelsea a lifeline midway through the second half, but the striker poked over.
McGoldrick put Chelsea out of their misery in the 77th minute.
He started the move on the halfway line and played a perfectly-weighted pass to Lys Mousset.
Mousset cut it back into the penalty area and when Rudiger made a complete hash of clearing, McGoldrick was on hand to blast into the roof of the net.