The Gunners' normally impenetrable defence was breached inside two minutes by the Blues, but they roared back for a 2-1 win to maintain a 100 per cent start.
City were less convincing in a 1-0 victory over Coventry, but the top two from last season are already clear at the top.
Manchester United are already five points back from the leaders after they twice blew the lead to draw at Everton.
AFP Sport looks at three talking points from the Premier League weekend.
Alonso's Chelsea handed title lesson
Chelsea arrived at the Emirates Stadium feted as a serious threat to Arsenal's hopes of retaining the title.
Over the following 90 minutes, Chelsea boss Xabi Alonso learnt the difference between pretenders and genuine contenders.
The Blues had kicked off Alonso's reign in free-scoring fashion, beating Fulham and Brighton in the top flight while netting seven times and easing past Luton in the League Cup.
When Morgan Rogers put Chelsea ahead with a stylish volley it appeared the Alonso revolution was in full swing.
But Arsenal are a far tougher proposition than Chelsea's first three opponents, as they proved with a stirring fightback.
Kai Havertz exposed the flaws in Alonso's three-man central defence as he drifted to the edge of the area for a low drive that eluded Emiliano Martinez's weak attempted save.
It was the kind of dubious keeping that Martinez was signed to eradicate after the hapless form of his predecessor Robert Sanchez.
Chelsea's vaunted attack of Rogers, Cole Palmer and Joao Pedro were largely held in check after their opening salvo by an Arsenal defence featuring a first start for Ezri Konsa.
The maturity and cutting edge of Mikel Arteta's men came to the fore again in the 50th minute when Martin Odegaard capped a flowing move with a clinical winner.
Alonso has plenty to work on, while Arsenal are in the rudest of health.
Isak shows sign of Liverpool life
Alexander Isak's three Premier League goals this season already matches his tally from an injury-disrupted first campaign at Liverpool.
The Swedish international's struggles after his English transfer record £125 million move from Newcastle played a major role in Liverpool's fall from champions to just scraping into the top five.
But the 26-year-old showed flashes of his old self at the World Cup and a full pre-season has allowed him to hit the ground running under Andoni Iraola.
An Isak double inside the first 10 minutes at Ipswich on Friday night secured a first win of the Iraola reign.
"I don't think the quality is anything that we doubt," said Liverpool captain Virgil Van Dijk.
"We've seen how good he is playing against him and now he's going to show the world with a Liverpool shirt as well."
No case for Man Utd defence
Everton fans arrived at the Hill Dickinson Stadium still in uproar at the lack of attacking options available to David Moyes after a chaotic final few hours of the transfer window saw a deal for Folarin Balogun fall through.
Yet the Toffees, just like newly-promoted Hull and Ipswich, still found a way to breach the Manchester United defence twice.
Ainsley Maitland-Niles's spectacular strike rescued a 2-2 draw for Moyes's men with virtually the last kick of the game, but it was little more than Everton deserved.
Excellent finishes from Bryan Mbeumo and substitute Benjamin Sesko and another encouraging performance from Marcus Rashford displayed United's strength going forward.
But the decision to not reinforce Michael Carrick's defence in the transfer window is already looking misguided before much tougher tests begin with the visit of Manchester City next weekend.