Everton enjoyed their first Premier League win at Manchester United for 12 years despite playing virtually the entire game with 10 men.
Everton's Ashley Young came on as a second-half substitute but his 18-year-old son Tyler remained on the Peterborough bench.
Chelsea, who beat Leicester City 2-0 at Stamford Bridge with goals from home debutant Diego Costa and Eden Hazard, and Swansea City, who sank Burnley 1-0 at the Liberty Stadium, head the early season standings with six points.
Liverpool survived a late spell of Everton pressure to stretch their record unbeaten run in Merseyside derbies to 17 games
Newcastle United striker Papiss Cisse has been charged with violent conduct, the FA said in a statement on Monday referring to an incident in the Premier League match against Everton.
Chelsea striker Diego Costa needs to learn to show respect to opponents after the Spaniard goaded Everton players during their pulsating 6-3 English Premier League win on Saturday, Toffees coach Roberto Martinez said.
Manchester United's top-four aspirations suffered a setback as Louis van Gaal's side were blown away 3-0 by Everton.
Arsenal snapped a season-long jinx against Newcastle United with a nervy 1-0 win in their last Premier League home game on Sunday.
Everton manager Ronald Koeman is not one to get carried away so it was unsurprising that he reacted to their best ever start to a Premier League season by focusing on where they needed to improve. Everton made light work of promoted Middlesbrough on Saturday, fighting back from going behind to win 3-1, with midfielder Gareth Barry marking his 600th Premier League appearance with a goal. Further efforts from Seamus Coleman and Romelu Lukaku wrapped up a victory that lifted unbeaten Everton back into second in the table after five games.
Everton's Romelu Lukaku has described himself as one of the most gifted strikers in world football after scoring 25 goals in all competitions for his Premier League club last season. Despite his form in front of goal Everton finished 11th and sacked manager Roberto Martinez, although Lukaku still rates the season as his finest in terms of individual achievement. "I'm among the best. That's good. Personally I am happy," Lukaku said in an interview with the June edition of World Soccer magazine. "I did expect to produce a season of this calibre." The 23-year-old is part of Belgium's squad for this month's European Championship in France, where his team play Italy, Ireland and Sweden in Group E.
Ireland reached the knockout phase at a European Championship for the firs time when Robbie Brady's late goal earned them a 1-0 victory against Group E winners Italy on Wednesday.
Liverpool will resume their quest for a first top-flight title in 30 years against Everton on June 21 but the venue for the behind-closed-doors fixture was not confirmed when the league released its revised fixtures for the season's restart.
Arsenal missed their chance to go back to the top of the Premier League after being forced to play with 10 men for much of the game against London rivals Chelsea on Sunday.
Tottenham Hotspur showed the English Premier League title race might not be just a two-horse affair with a hugely impressive 6-2 win at Everton.
A summary of Thursday's action in the Premier League.
'We always knew it was going to be an important one playing against a quality side, but I think it affected us a little bit too much.
Everton were guilty of celebrating Ross Barkley's late goal for too long as Bournemouth snatched a dramatic 3-3 draw in the eighth minute of stoppage time on Saturday, goalkeeper Tim Howard said.
Swansea City equalled their highest ever points tally in a Premier League season following a 1-1 draw with Everton in the early game at the Liberty Stadium on Saturday.
Sadio Mane struck four minutes into stoppage time to give Liverpool a last-gasp 1-0 win over derby rivals Everton that lifted them into second place in the English Premier League.
Louis van Gaal accused his Manchester United players of being out-fought in a 3-0 defeat at Everton.
Images from Saturday's Premier League action.
IMAGES from the Premier League matches played on Saturday.
IMAGES from the EPL matches played on Saturday
Manchester United maintained their push for a top-four Premier League finish when they beat Everton 1-0 with a second-half goal by French striker Anthony Martial at Old Trafford on Sunday.
IMAGES from the Premier League matches played on Saturday.
A Ciaran Clark own goal earned Sweden a fortunate 1-1 draw against a spirited Ireland side in their Euro 2016 Group E opener at the Stade de France on Monday.
Bruno Fernandes struck from the penalty spot in the ninth minute of stoppage time to give Manchester United victory at Brighton & Hove Albion in the Premier League on Saturday after an incredible finale.
Photos from the English Premier League matches played on Wednesday
Everton secured their English Premier League status as Dominic Calvert-Lewin scored five minutes from time to snatch a thrilling 3-2 victory over Crystal Palace after they rallied from two goals down at a rocking Goodison Park on Thursday.
Manchester United have enough in their armoury to cope with the loss of suspended striker Wayne Rooney, Everton manager Roberto Martinez warned his players on Friday ahead of this weekend's Premier League clash at Old Trafford.
It promises to be a tense and thrilling finale to the season but Liverpool, who have frittered away a seven-point lead, will need to regain their momentum quickly if they are to push for their first league title since 1990.
Ashley Williams powered home a towering header in the 86th minute as Everton fought back to claim a 2-1 win over Arsenal as the title hopefuls blew the chance to return to the Premier League summit.
With Manchester City drawing 2-2 at Arsenal and Liverpool suffering a shock 1-0 home defeat by Aston Villa, Jose Mourinho's Chelsea were the big winners in the fourth round of Premier League games.
IMAGES from the English Premier League matches played on Sunday
News of all that's transpired on and off the football field.
The injury to the Chile forward, who was left out of the squad for Tuesday's Champions League victory over Young Boys, "will require further investigation to confirm its severity", the club added.
It was a day of role reversal as Jose Mourinho's often cautious Chelsea thumped Everton 6-3 in a thriller on Saturday while champions Manchester City's attacking talents were blunted in a shock 1-0 home defeat by Stoke City.
A little over six years after almost losing his life in a training-ground accident with the Ireland team, defender Shane Duffy will probably start one of the biggest games of his career against France in the second round of Euro 2016.
Bournemouth breathed life into their Premier League survival bid on Sunday, coming from a goal down to beat fourth-placed Leicester City 4-1 for their first win in 10 games. Toby Alderweireld's 81st-minute header sealed a 2-1 Premier League victory for Tottenham Hotspur against Arsenal in the first north London derby in their new stadium on Sunday.
Leicester beat Southampton to open seven-point gap at the top.