Liverpool won the Merseyside derby 2-0 on Sunday to move back to within a point of Premier League leaders Manchester City and push neighbours Everton into the relegation zone.
Everton's troubles were enhanced by a rejuvenated Burnley beating Wolverhampton Wanders 1-0 at Turf Moor to move out of the bottom three for the first time since October.
Chelsea kept a grip on third place but needed a last-minute goal from American Christian Pulisic to beat West Ham United 1-0 at Stamford Bridge.
City, who beat lowly Watford 5-1 on Saturday, have 80 points from 33 games with Liverpool on 79 from the same number of matches. Chelsea are third on 65 with Arsenal in fourth spot on 60.
Liverpool found it hard going against a determined Everton at Anfield but it was derby specialist Divock Origi, usually a fringe player, who turned the game in favour of Juergen Klopp's side.
Origi was brought on in the 60th minute along with Colombian winger Luis Diaz and Klopp's switch paid instant dividends.
Two minutes after the change, Belgian forward Origi linked up with Mohamed Salah before the Egyptian floated a cross to the back post which was headed in by full-back Andy Robertson.
Then five minutes from the end, Origi made sure of the win heading in after a spectacular attempted volley from Luis Diaz bounced to him at the back post.
Origi has now scored six times in nine games against Everton in the Premier League.
"It is a nice record. When I go on the pitch, I just think about making plays and playing football. Today we needed it. It was a tough game; we knew it was going to be tough for the last games. Eventually we got the three points, so that’s the most important thing," he said.
Everton face a nervy end to the season as they look to keep their record of never having been relegated since the creation of the Premier League.
Lampard's side are now in 18th place on 29 points from 32 games while Burnley are two points ahead having played one game more.
The Clarets have now earned seven points from the three games since they sacked long-term manager Sean Dyche with caretaker Mike Jackson giving them renewed hope of survival.
A 62nd-minute goal from Czech forward Matej Vydra, who converted a low cross from Dutchman Wout Weghorst, was enough for Burnley against eighth-placed Wolves.
Chelsea ended a run of back-to-back home league defeats after substitute Pulisic arrived in the box at the right time to guide home a pass by Marcos Alonso in the 90th minute, shortly after Jorginho had seen a penalty saved by West Ham keeper Lukasz Fabianski.
West Ham defender Craig Dawson was shown a red card after a check by the Video Assistant Referee following a clumsy foul on substitute Romelu Lukaku which resulted in the penalty.
Stunning long-range goals in each half from captain James Ward-Prowse helped Southampton rescue a point at Brighton & Hove Albion with a 2-2 draw at the Amex Stadium.
Graham Potter's side were cruising thanks to an early strike from Danny Welbeck and an own goal by Mohammed Salisu, but Ward-Prowse sparked a comeback after scoring directly from a free kick and completed it with a fierce low drive from 22 yards.