London clubs' erratic start to the Premier League season improved with victories for Chelsea, Tottenham Hotspur and Arsenal on Saturday but Manchester sides continued to make the running, with City and United the top two.
Chelsea's 2-1 win away to Leicester, in the meeting of the previous two seasons' champions, was the tightest.
N'Golo Kante scored against his former club soon after halftime to extend Chelsea's lead after Alvaro Morata claimed his third goal in four league games since moving from Real Madrid.
Leicester, having made two changes at the interval, halved the deficit with a Jamie Vardy penalty but the champions held on to move into third place.
Tottenham's Harry Kane reached 100 goals for the club with a freak opening effort in the comfortable 3-0 win at Everton, adding a second later.
After the home side started brightly, the England striker deceived goalkeeper Jordan Pickford from way out on the right wing just before the half-hour.
"I got a lucky one but sometimes you need that," he said
"It has been a crazy few years and to get 100th goal was amazing."
Christian Eriksen scored the second after Pickford made a good save from Ben Davies and Kane knocked in Davies' pass a minute after the interval.
The win pushed Spurs into the top five and provided a boost ahead of their opening Champions League game against Borussia Dortmund at Wembley, where they are playing home league games this season but have yet to win one.
Their north London rivals Arsenal, heavily criticised after defeats by Stoke City and Liverpool, were easy winners at home to struggling Bournemouth, prevailing 3-0.
Danny Welbeck gave them the lead and close-season signing Alexandre Lacazette doubled it before halftime. After Jermain Defoe headed against a post with the visitors' first attempt on goal, Arsenal broke out in characteristic fashion for Welbeck to claim his second.
He was close to a hat-trick as Bournemouth were pushed into the bottom two on goal difference by their fourth successive defeat.
At the other end of the table Manchester City went top by a point ahead of Manchester United's late game at Stoke, trouncing Liverpool 5-0 after Juergen Klopp's side had Sadio Mane sent off in the 37th minute.
He was adjudged guilty of endangering an opponent by catching Brazilian goalkeeper Ederson in the face with a high foot.
Already trailing to a Sergio Aguero goal, Liverpool conceded two more to Gabriel Jesus on either side of halftime and suffered a rare heavy defeat when Leroy Sane also scored twice in the final quarter of an hour.
Ederson was taken off on a stretcher but tweeted later that he was fine.
"Hi, everyone - I am well," he said. "What a fright! It was an important win for us. We are strong. Thanks for all your messages of support."
Promoted Brighton, back in the top tier for the first time since 1983, achieved their first win, by 3-1, at home to West Bromwich Albion.
New signing Pascal Gross scored twice and Tomer Hemed added the third before a consolation by James Morrison, giving West Brom a first defeat.
In the day's other game Watford moved above Southampton and into the top four by beating them 2-0 with goals by Abdoulaye Doucoure and substitute Daryl Janmaat.
Bottom club Crystal Palace play at Burnley and Swansea City host Newcastle United on Sunday, while West Ham United who, like Palace, have lost their first three games, face Huddersfield Town on Monday.
