Match ends, Everton 1, Manchester City 3.
Manchester City had to work hard to see off a spirited Everton side and stay five points behind Premier League leaders Liverpool.
Riyad Mahrez and Raheem Sterling both struck in the final 20 minutes to give City victory after they had wasted some good chances, but they needed two excellent saves by Ederson to hold off the Toffees.
It was a convincing end to a tense evening, with Pep Guardiola's side going to a murky Merseyside knowing only victory would suffice after seeing the Reds maintain their flawless start earlier on Saturday.
The defending champions started well and, after Ilkay Gundogan somehow hit the bar from point-blank range, they took a deserved lead thanks to another moment of genius from Kevin De Bruyne.
His trademark whipped cross from the right was nodded home by Gabriel Jesus, giving the Belgian his eighth assist of a stellar campaign.
That spelled trouble for Everton, whose boss Marco Silva was under pressure before kick-off after back-to-back league defeats, but they responded well and were soon level.
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Toffees substitute Alex Iwobi fought his way into the box and the ball dropped for Seamus Coleman, who beat Ederson with a dinked shot before Dominic Calvert-Lewin made sure of the equaliser with a diving header on the line.
City were playing some neat football but with top scorer Sergio Aguero starting on the bench along with David Silva and Bernardo Silva, they could not make it count in the final third.
Sterling should have restored their lead at the start of the second half when he ran clear from a delightful Mahrez pass, but his first-time finish fizzed wide.
Everton were asking questions too, particularly from set-pieces, before Mahrez provided the spark City had been waiting for when he curled home a stunning free-kick into the bottom corner from 20 yards out.
The home side refused to lie down and Ederson made vital stops from Calvert-Lewin then Yerry Mina, but Sterling made sure of the points when he forced the ball home after Aguero's cross-shot was saved.
Everton deserve plaudits despite defeat
Everton were booed off the Goodison Park pitch after losing here to Sheffield United seven days ago but they deserve some plaudits for responding with a much-improved performance this time.
Yes, it was still a fourth defeat in their past five league games, but they played with passion, passed with purpose - and were close to claiming a surprise point.
A lack of cohesion in the Toffees attack despite a £125m summer spending spree has been one of the reasons Silva's job is supposedly under threat, but they gave City's defence a thorough test.
That was despite losing Theo Walcott to concussion in the first minute of his first start of the season, with the former England forward unable to continue after being hit in the face by a Raheem Sterling cross.
His replacement, Iwobi, was one of those who impressed, along with the industrious Richarlison and Fabien Delph, a calming influence in midfield against his old club.
There were spells in the second half with the score at 1-1 when it appeared Silva might finally lay to rest one of the statistics that haunts him here - his team's inability to ever come back from behind to win in a league game.
That feat eluded him for the 22nd time in his reign as Toffees boss, but this was still a display that offered some encouragement.
City's defence has a difficult evening
Since being beaten at Norwich and suffering their first defeat of the season, City had enjoyed three straight wins in three different competitions, scoring 14 goals and conceding none.
This, however, was a far more difficult day for their defence as Everton's previously shot-shy front-line offered a far sterner test than Shakhtar Donetsk, Watford or Preston had managed.
And, while City survived and left with the three points, that was more down to Ederson's brilliance than his creaking back-line, which was exposed again - particularly from set-pieces.
A succession of corners and free-kicks from the right caused them problems in the second half, with Richarlison heading wastefully over at the near post from one, and Mina's bouncing effort bringing the best out of Ederson soon after.
Worryingly for City, Everton were finding a way through them in open play too, and the result might have been very different had Ederson not rushed from his line to deny Calvert-Lewin with the score at 2-1.
Man of the match - Riyad Mahrez (Man City)
'Positives despite defeat' - what they said
Everton boss Marco Silva: "The first 20 minutes our performance wasn't good enough but after their goal we responded strong. We had some good moments. We were faster and quicker and made problems for them and equalised.
"We created problems for them and had a good moment with Gylfi Sigurdsson. They had a bit more possession, which is normal against most teams.
"But we had a good attitude from our players and, to be honest, we had the best chances before they scored to make it 2-1, through Mina and Calvert-Lewin. There was a big save by Ederson.
"It was a good game, a competitive game. We didn't get the points but there were lots of positives."
Man City boss Pep Guardiola: "At Goodison Park it is always difficult, at this stadium but we started incredibly well as usual and we created chances and scored a goal.
"But we created from wide areas in the second half and we had the quality of Riyad Mahrez for the goal.
"Everton have fast players, they caused a lot of problems at the back, so I'm delighted because we want to go for this league and try to win it again."
City feast on Toffees again - the stats
- Everton have won just one of their last 13 Premier League matches against Manchester City (W1 D4 L8).
- Manchester City have scored 27 goals in their opening seven league games this season - the most by a top-flight team in the opening seven matches of a season since Everton in 1894-95 (27).
- Everton directed eight shots on target against Manchester City, the most any side has managed in a Premier League clash with the Citizens since November 2015, when Southampton also fired in eight.
- Gabriel Jesus has scored 22 goals in his last 21 starts for club and country, netting in each of his last six starts for Manchester City.
- Kevin De Bruyne has provided eight assists in the Premier League already this season - only Cesc Fabregas in 2009-10 has provided as many in his side's first seven games in a single campaign.
- Everton striker Dominic Calvert-Lewin has scored in back-to-back Premier League appearances for the first time in his career, and in back-to-back league games for the first time since October 2015 whilst playing for Northampton in League Two.
- Riyad Mahrez has been directly involved in 17 goals in his last 17 starts for Manchester City in all competitions (seven goals, 10 assists), including nine in his last seven (four goals, five assists).
- Manchester City winger Raheem Sterling scored his 100th goal in all club competitions, with 77 coming for the Citizens and 23 for Liverpool.
- Manchester City's victory over Everton earned David Silva his 200th Premier League win in his 289th game in the competition, making him the quickest ever player to reach this total.
City have two home games coming up, on Tuesday 1 October against Dinamo Zagreb in the Champions League (20:00 BST) and then on Sunday 6 October against Wolves in the Premier League (14:00 BST).
Everton hit the road next weekend, travelling to Burnley on Saturday (15:00 BST).
- 13MinaBooked at 70mins
- 18SchneiderlinBooked at 59minsSubstituted forDaviesat 81'minutes
- 11WalcottSubstituted forIwobiat 7'minutesSubstituted forKeanat 74'minutes
- 10G Sigurdsson
- 30OtamendiBooked at 51mins
- 17De BruyneSubstituted forBernardo Silvaat 80'minutes
- 16RodriBooked at 38mins
- 9Gabriel JesusSubstituted forAgüeroat 66'minutes
- 7SterlingSubstituted forSilvaat 86'minutes
- 20Bernardo Silva
- Michael Oliver
- Shots on Target
Second Half ends, Everton 1, Manchester City 3.
Offside, Manchester City. David Silva tries a through ball, but Bernardo Silva is caught offside.
Corner, Everton. Conceded by Oleksandr Zinchenko.
Foul by Rodrigo (Manchester City).
Tom Davies (Everton) wins a free kick in the defensive half.
Substitution, Manchester City. David Silva replaces Raheem Sterling.
Foul by Kyle Walker (Manchester City).
Richarlison (Everton) wins a free kick in the defensive half.
Goal! Everton 1, Manchester City 3. Raheem Sterling (Manchester City) left footed shot from the centre of the box to the bottom left corner.
Attempt missed. Sergio Agüero (Manchester City) right footed shot from a difficult angle on the right misses to the left. Assisted by Riyad Mahrez.
Attempt saved. Yerry Mina (Everton) header from the centre of the box is saved in the centre of the goal. Assisted by Gylfi Sigurdsson with a cross.
Corner, Everton. Conceded by Raheem Sterling.
Substitution, Everton. Tom Davies replaces Morgan Schneiderlin.
Substitution, Manchester City. Bernardo Silva replaces Kevin De Bruyne.
Attempt missed. Dominic Calvert-Lewin (Everton) left footed shot from the left side of the six yard box is close, but misses to the left following a corner.
Attempt saved. Yerry Mina (Everton) header from the centre of the box is saved in the bottom left corner. Assisted by Gylfi Sigurdsson with a cross.
Corner, Everton. Conceded by Oleksandr Zinchenko.
Corner, Everton. Conceded by Ederson.
Attempt saved. Dominic Calvert-Lewin (Everton) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Gylfi Sigurdsson with a through ball.