The champions beat Aston Villa 2-1 while Manchester United went down 1-2 at home to Blackburn Rovers in the Premier League.
Chelsea and AC Milan drew 1-1 in a friendly match on Sunday as they wrapped up their pre-season tour of the United States.
The England midfielder is confident of filling a defensive midfield role, formerly taken by Nicky Butt.
The Chelsea midfielder said he was sad to see coach Claudio Ranieri go.
The English champions-elect, 4-2 up from the first leg, went through 6-5 on aggregate in the Champions League.
A round-up of Saturday's action in the Premier League.
An 85th minute goal from a free kick by Ireland winger Damien Duff gave Chelsea a 2-1 victory over Manchester United.
Images from the English Premier League matches played on Saturday
Frank Lampard's loan spell at Manchester City is due to end in January but the Premier League champions might want to delay his departure across the Atlantic to enhance their chances of fighting his old club for the title.
The 1-0 victory gave Chelsea top slot in the premier league, ahead of Arsenal.
Chelsea booked their place in the League Cup semi-finals with a late 2-1 win at Fulham.
Manchester City moved 17 points clear of the pack with a 3-0 win at Fulham on Saturday
"He has still got a lot to learn, he has to improve when he is part of a defensive organisation," the Chelsea coach said.
IMAGES from the English Premier League match played between Arsenal and Chelsea at Emirates Stadium in London on Sunday.
The 26-year-old midfielder signed a new five-year contract with the English premier league club.
Chelsea's third successive league win lifted them level on 18 points with Leicester City but are top thanks to a better goal difference. Leicester face Liverpool on Sunday.
Chelsea's 0-0 home draw with second-placed United ensured Thomas Tuchel is still undefeated in nine games since replacing Frank Lampard, but it meant they failed to dislodge fourth-placed West Ham United who became Manchester City's 20th successive victims in all competitions on Saturday.
It rallied from a 2-0 deficit to draw 2-2 with Chelsea and reach the Champions League final 5-3 on aggregate.
Newly-formed New York City FC will begin their preparations for their Major League Soccer (MLS) debut with two friendly matches in Manchester next month, Manchester City said on Tuesday.
Images from the English Premier League matches played on Wednesday night.
IMAGES from the English Premier League matches played on Sunday.
Chelsea manager Frank Lampard was delighted with the discipline of his team, who ended United's 19-game unbeaten run.
Five Manchester City first team players have tested positive for COVID-19 and will not be involved when they play Chelsea in the Premier League this weekend
Manchester United dropped out of the Premier League's top four
Manchester United completed their first home and away Premier League double over Manchester City since the Alex Ferguson era with a 2-0 victory at Old Trafford on Sunday.. Earlier, two goals in each half from a rampant Chelsea earned them a 4-0 victory over Carlo Ancelotti's Everton, cementing their place in the Premier League's top four.
The Super Cup is an annual match played between the winners of the Champions League and the Europa League
Liverpool are champions and Norwich City, the side whom they beat on Aug. 9 to launch their title charge, are down and Manchester City are distant runners-up. After that everything is up for grabs.
After a close season in which he said he wanted to leave the club but ended up staying at Old Trafford, all eyes were on Paul Pogba as Manchester United began their Premier League campaign on Sunday. Yet following a 4-0 home win over Chelsea, in which the French midfielder set up the third and fourth goals, manager Ole Gunnar Solskjaer was full of praise for the former Juventus star.
Messi, whose deal expires in 2021, had called his side "weak" and "inconsistent" after they lost the league title to Real Madrid.
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