Liverpool secured a dominant win over West Ham, while other Premier League matches saw Brentford edge Burnley in a seven-goal thriller and Everton secure a late victory against Newcastle.
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Eberechi Eze timed his return to the Arsenal scoresheet to perfection with two goals in a 4-1 derby rout of Tottenham Hotspur.
City, Liverpool all cruise in respective FA Cup matches played on Saturday.
Liverpool coach Arne Slot said he was delighted to have Mohamed Salah returning from Africa Cup of Nations duty soon, despite an apparent falling-out just weeks ago.
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Images from the EPL matches played on Saturday.
Carrick faces first test as United boss in Manchester derby showdown
Arne Slot is the first to win the top flight in England where many of his compatriots have suffered dents to their reputations over the years.
Chelsea battled from two goals down to beat West Ham 3-2 in the Premier League on Saturday with a stoppage-time Enzo Fernandez winner.
Liverpool forward Mohamed Salah is destroying his legacy at the club with his fiery reaction to being left out of the side, former Manchester United striker Wayne Rooney said.
Arne Slot planned to have a face-to-face meeting with Mohamed Salah on Friday ahead of Saturday's Premier League match against Brighton & Hove Albion.
IMAGES from UEFA Champions League matches played on Wednesday
Liverpool striker Alexander Isak is expected to be sidelined for at least a couple of months after breaking his leg and undergoing surgery.
Second-placed City would have reduced Arsenal's advantage to two points but instead were frustrated as their winning streak in all competitions was halted at eight.
Liverpool could be 14th in the table by the time they kick off at West Ham on Sunday afternoon and they will be desperate to steady the ship with three points.
Liverpool manager Arne Slot must find a way to arrest the team's slide even as the pressure mounts on him, former skipper Steven Gerrard said.
IMAGES from the Premier League matches played on Sunday
The Egypt forward has been left out of the club's Champions League match against Inter Milan.
After Saturday's 2-1 win at Chelsea - their 12th in 13 league matches - Aston Villa have claimed 18 points from losing positions so far this season, more than any other team.
Premier League leaders Arsenal needed two own goals, the second deep in stoppage time, to scrape a 2-1 home win over bottom club Wolverhampton Wanderers and open a five-point gap, while earlier on Saturday Mohamed Salah put aside a tumultuous week to help Liverpool beat Brighton and Hove Albion 2-0.
Carragher said manager Arne Slot should build the team around Alexander Isak and Florian Wirtz.
While his overall win rate is more than 70%, Liverpool have been a tough nut for Pep Guardiola to crack since he arrived at City in 2016, first under Juergen Klopp and now Arne Slot.
Arsenal missed a chance to tighten their grip on the Premier League title race with a 1-1 draw at 10-man Chelsea on Sunday.
Liverpool shrugged off the absence of Mohamed Salah as midfielder Dominik Szoboszlai scored an 88th-minute penalty to give them a 1-0 Champions League win at Inter Milan.
Goals from Pedro Neto and Enzo Fernandez earned Chelsea a 2-0 victory at lowly Burnley on Saturday, a win that moved the Blues within three points of Premier League leaders Arsenal.
Second-string Liverpool were thumped 3-0 by Crystal Palace as Ismaila Sarr struck twice for the visitors in the League Cup on Wednesday, booking the London side's spot in the quarter-finals and piling more misery on Arne Slot's struggling side.
Arsenal maintained a searing pace at the top of the Premier League with a straightforward 2-0 win at Burnley, while Liverpool snapped a dreadful four-match losing run with a 2-0 victory over Aston Villa to climb provisionally to third.
IMAGES from the EPL matches played on Saturday
Manchester United manager Ruben Amorim would have enjoyed some relative calm during the international break after backing from club co-owner Jim Ratcliffe but will find himself in the Premier League maelstrom once again at Liverpool on Sunday.
Bayern Munich made it 16 wins from 16 games this season to underline their credentials as early Champions League favourites, beating holders Paris St Germain 2-1 away as Luis Diaz scored two goals and was shown a red card on Tuesday.
IMAGES from the UEFA Champions League matches played on Wednesday
Virgil van Dijk scored with a stunning late header in the second minute of stoppage time to give Liverpool a 3-2 victory over Atletico Madrid in their opening group stage match of the Champions League, at Anfield, on Wednesday.
Nine players from treble-winning Paris St Germain have been nominated for the men's Ballon d'Or this year along with the club's coach Luis Enrique as France Football announced the list of nominees on Thursday.
Liverpool's Dominik Szoboszlai struck the first telling blow in the Premier League title race with a dazzling free kick late on giving his side a 1-0 victory over rivals Arsenal.
Reigning Premier League champions Liverpool honour Portugal forward Diogo Jota, who died in a car crash alongside his brother Andre Silva last month, ahead of the season-opener kick-off against Bournemouth on Friday.
Carrying a red floral wreath bearing his shirt number, Diogo Jota's Liverpool teammates joined relatives and residents in a small Portuguese town on Saturday for the funeral of the soccer star.
Liverpool talisman Mo Salah and substitute Federico Chiesa struck late goals as the Premier League champions began the defence of their title by beating Bournemouth 4-2 in a thriller on the opening night of the season at Anfield on Friday.
Crystal Palace beat Premier League champions Liverpool 3-2 on penalties to win the Community Shield after an entertaining 2-2 draw at Wembley on Sunday.
Champions Liverpool edged Newcastle United 3-2 in a thrilling finish thanks to a stoppage-time winner by teenaged Premier League debutant Rio Ngumoha.