*Draws for Chelsea, Saints, Everton, Liverpool
Raheem Sterling scored the winner as Manchester City edged past Arsenal to climb up to 10th in the Premier League.
England international Sterling finished off a slick team move by drilling in the only goal on 23 minutes.
Arsenal had chances to get back into the game but City goalkeeper Ederson spread himself big to deny both Bukayo Saka and Pierre-Emerick Aubameyang.
Thomas Partey came on for his Arsenal debut in the second half but the Gunners could not find an equaliser.
At Stamford Bridge, Frank Lampard said Chelsea need to cut out individual errors after they twice relinquished a lead to draw 3-3 with Southampton.
Jannik Vestergaard headed in a dramatic injury-time equaliser to claim the Saints a point, after Danny Ings and Che Adams had earlier brought the visitors back from 2-0 down to 2-2 at Stamford Bridge.
All the goals conceded by the Blues included defensive mistakes, with Kai Havertz needlessly losing possession in his own half for the first, a weak back-pass from Kurt Zouma and poor attempted block from goalkeeper Kepa Arrizabalaga contributing to the second, and an unnecessarily conceded free-kick and poor attempted clearance enabling the third.
“We have to not give those goals away,” Lampard told BBC Sport. “We are making individual mistakes in games and it gives other teams chances at all times. It is something we need to get better at.
“It goes through the team. We are seeing a lot of goals in the Premier League at the moment and no manager is happy with that. We all want clean sheets. It has to be better.”
Chelsea were in control courtesy of two superb Timo Werner goals – the first courtesy of a sublime dummy which bamboozled Jan Bednarek before drilling in, and the second as he lobbed the ball over goalkeeper Alex McCarthy and nodded home.
When it looked like the hosts were cruising, Southampton hit back as Danny Ings went round keeper Arrizabalaga to finish and Adams punished dire Chelsea defending to smash home an equaliser.
In the first game of the day,Everton came from behind twice to maintain their unbeaten start to the Premier League in a Merseyside derby thriller as Liverpool were denied a stoppage-time winner by the video assistant referee.
Sadio Mane put the visitors ahead in the third minute but they were furious soon afterwards when a wild challenge by Everton keeper Jordan Pickford on Virgil van Dijk, committed after an offside decision, saw the defender having to go off injured with VAR failing to review the tackle.
Everton, who had won their previous seven games in all competitions, equalised swiftly at Goodison Park when Michael Keane headed in a corner from the outstanding James Rodriguez.
Liverpool, with Thiago Alcantara at the heart of everything, were back in front with 18 minutes left when Mohamed Salah volleyed in Yerry Mina’s poor clearance to set up a frantic finale, with only Pickford’s magnificent save from Joel Matip’s header keeping Everton in the hunt.
Dominic Calvert-Lewin’s towering header restored parity with nine minutes left but Everton were then reduced to 10 men when Richarlison was sent off for a wild challenge on Thiago.
Liverpool thought they had won it in stoppage time when Pickford could not keep out Jordan Henderson’s shot – but in a finish to match the chaotic nature of the game, Mane was adjudged to be offside after a lengthy wait for the VAR decision.
Everton 2 – 2 Liverpool
Chelsea 3 – 3 Southampton
Man City 1 – 0 Arsenal