West Ham's 100 per cent start was halted in a pulsating derby clash with Crystal Palace, which ended 2-2 at the London Stadium.
Palace midfielder Conor Gallagher, on a season-long loan from Chelsea, twice equalised for Patrick Vieira's side after Pablo Fornals and then Michail Antonio's 50th Premier League goal had edged the Hammers in front.
Everton extended their unbeaten start under Rafael Benitez with a deserved 2-0 win at Brighton.
Summer signing Demarai Gray's second league goal of the season gave Everton an early lead and Dominic Calvert-Lewin converted a 58th-minute penalty after Joel Veltman's foul on Seamus Coleman.
It was Everton's first win at Brighton since 1983 and ended the Seagulls' 100 per cent start to the season.
Leicester bounced back from Monday night's 4-1 defeat at West Ham with a 2-1 win at Norwich, who remain without a point.
Jamie Vardy pounced on a mistake by Norwich's on-loan Manchester United defender Brandon Williams to give the Foxes an early lead.
Teemu Pukki equalised from the penalty spot after Caglar Soyuncu's sliding tackle on Pierre Lees-Melou, but the visitors regained the lead through Marc Albrighton's deflected shot.
Norwich were denied a second equaliser by VAR when Kenny McLean headed in a corner, with Todd Cantwell deemed to have been offside.
Brentford remain unbeaten since winning promotion to the top flight following a 1-1 draw at Aston Villa.
Ivan Toney's thumping finish gave Brentford an early lead before Villa's summer signing Emiliano Buendia levelled five minutes later with his first goal for the club, a superb finish into the top corner.
Newcastle registered their first point of the season, but were denied victory by James Ward-Prowse's stoppage-time penalty in a thrilling 2-2 home draw against Southampton.
Ward-Prowse converted from the spot in the sixth minute of added time after VAR had spotted Jamaal Lascelles' foul on Adam Armstrong.
Newcastle thought Allan Saint-Maximin had grabbed them a stoppage-time winner after Callum Wilson's second-half header had been cancelled out by Mohamed Elyounoussi.