Liverpool strolled to a 2-0 victory over Brentford, while Newcastle drew 1-1 at Bournemouth.
The 51-year-old resigned from Blackburn Rovers where he had been in charge since 2000.
Liverpool and Manchester City having the potential to "dominate" European football, former Scotland international Graeme Souness has said.
Former Scotland midfielder Graeme Souness said FIFA needed to be more transparent over VAR decisions.
Manager Graeme Souness' dismissal comes following the club's poor run of form this season.
Another Liverpool great, Graeme Souness, said Klopp had produced a "fabulous" side.
The Newcastle United manager is confident striker Alan Shearer will put off retirement plans and stay on for one more season with the club.
Alan Shearer broke Newcastle United's goal-scoring record in a 2-0 home win over fellow strugglers Portsmouth.
Captain Thierry Henry converted a late penalty to give Arsenal a 2-0 win over 10-man Newcastle United
Alan Shearer said he is not ready to become manager of Newcastle United after the sacking of Graeme Souness
The Newcastle United striker has postponed his retirement and will play on next season on a player/coach contract.
Alan Shearer, set to hang up his boots at the end of this season, was persuaded to stay by Newcastle boss Graeme Souness
The 17-year-old midfielder, also called Diego Maradona, is currently at his father's former club Napoli in Italy's Serie B.
Here is some reaction to the return of the Portuguese manager to club soccer after being dismissed by Manchester United last December:
"...This club is built on what they did. It's built ... on (Bill) Shankly, (Bob) Paisley and (Joe) Fagan and all the others, but on these players... Steven Gerrard -- this club's built on, in the last 20 years, on Stevie's legs..."
first English title for 30 years had been sealed almost a month ago with seven games remaining, following a three-month stoppage because of the COVID-19 pandemic.
Pogba also said he is no longer bothered by criticism from soccer pundits because he has grown used to it.
Liverpool raised the bar so high and enjoyed such a huge winning margin in claiming the English Premier League title last season it would be churlish to doubt their credentials for a repeat showing.
Reactions to Liverpool's Premier League triumph, their first league title since 1990:
'I think they're in a big fight now'
Arsene Wenger remained defiant on Sunday after Arsenal's 3-2 capitulation at Manchester United prompted much criticism, with one distinguished former international suggesting their performance bordered on 'a joke'.
Twelve of Europe's leading soccer clubs from England, Italy and Spain announced a breakaway league on Sunday but after 48 hours of massive pressure from fans, politicians and even British royals the six English clubs backed out
News of all that's transpired on and off the football field
Van Dijk was substituted early in the game when Everton goalkeeper Jordan Pickford made a reckless challenge in the box, which left Liverpool manager Juergen Klopp fuming.
'I thought we need someone to sit in front of the defence who is cool and can be a third defender, but who can also play on the ball. I am very proud of him.'
Manchester City goalkeeper Joe Hart generated some more negative headlines after a last minute mix-up with Matija Nastasic allowed Chelsea's Fernando Torres to score a late winner in a 2-1 victory at Stamford Bridge on Sunday.
'You cannot sack all the players so sack the manager -- whether right or wrong'
Many former players believe that storming out of games is a powerful tool to counter racism, a move that will force administrators to take the problem more seriously, and act in a way that compels clubs to obliterate such a culture and evict those responsible, says Dhruv Munjal.
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