Third-placed Brighton ended their wait for a first ever away win against City at the 18th attempt to bolster dreams of challenging for the title.
The Seagulls have never won a major trophy but have already beaten champions Arsenal convincingly this season to show their title credentials.
Fabian Hurzeler’s men got revenge for defeat to City on penalties in the League Cup on Wednesday.
Hurzeler felt aggrieved at that result after his side twice led at the Etihad and this time held on after Yasin Ayari slotted them into an 11th-minute lead.
Bart Verbruggen was inspired in the Brighton goal to see off a City onslaught after half-time.
But the visitors were well worthy of a famous win and could also have added to their lead when Jack Hinshelwood hit the side-netting late on.
City won their opening seven league games as Enzo Maresca made a flawless start to the daunting task of succeeding Pep Guardiola.
However, the 10-time English champions know their results on the field may count for little this season unless they can successfully overturn a guilty verdict on over 100 charges of breaking Premier League financial rules.
City have appealed, but if that ruling is upheld they are likely to face a severe points penalty and possible relegation from the English top-flight.
Defeat for Maresca’s men means Arsenal can go top of the table if they avoid losing at Liverpool on Sunday.
Manchester United’s nine-game winless run is their longest without victory for 38 years after a 2-0 defeat at Chelsea.
The Red Devils were left to regret not taking advantage of a series of first half chances to take the lead before they imploded in the second period.
Morgan Rogers was allowed to run straight through the heart of the United defence to open the scoring.
Lisandro Martinez then did his under-pressure manager no favours with two quickfire yellow cards, the second of which also conceded a penalty.
Cole Palmer converted from the spot to lift Chelsea into the top four.
“The two halves of football couldn’t probably be further apart,” said Carrick, who added he expects to be in charge for Tuesday’s Champions League tie against Roma.
“We looked in a good place going into half-time and then found a way of unravelling it second half.”
Only Nottingham Forest’s 2-0 defeat at Brentford ensured United did not drop into the bottom three.
Nikola Milenkovic’s own goal and Igor Thiago’s strike took the Bees up to sixth.
Bournemouth extended their fine form with a 2-0 win over Leeds thanks to goals from Evanilson and Amine Adli.
Sunderland remain rooted to the foot of the table after a 3-1 defeat at Coventry.
Hull and Ipswich played out a 2-2 draw. (AFP)
