Liverpool took advantage of an early-season slip-up by Manchester City in a 1-1 draw at Wolves to move top of the Premier League as Mohamed Salah's winner saw off Brighton at Anfield on Saturday.
At the Emirates, Arsenal ended their wait for a victory under new boss Unai Emery by coming from behind to beat West Ham 3-1.
But Everton's flying start to their new era under Marco Silva stumbled as they threw away a 2-0 lead to draw 2-2 at Bournemouth and saw �50-million record signing Richarlison sent off.
City looked in ominous form to start the campaign in cruising to victory against Arsenal at the Emirates and thrashing Huddersfield 6-1 last weekend.
But a mixture of an inspired performance from newly-promoted Wolves and some misfortune for Pep Guardiola's men denied the visitors at Molineux as Willy Boly controversially opened the scoring by turning Joao Moutinho's cross into the net with his arm.
City responded via an unexpected source 21 minutes from time when Aymeric Laporte's towering header registered his first goal for the club.
Raheem Sterling and Sergio Aguero struck the woodwork for the visitors before the break and Aguero was denied again by the crossbar with virtually the last kick of the game from a fiercely dipping free-kick.
However, City boss Pep Guardiola insisted he had no regrets and even claimed it had been a "good point" for his side.
"We created enough chances to win the game but it doesn't count and we drew the game 1-1 and I don't have too many regrets," said Guardiola.
"There was a little lack of rhythm sometimes but I am satisfied in general with the performance."
- Salah sees off Brighton -
Liverpool were also far from their flowing best, but had enough to see off a resolute Brighton 1-0 thanks to one moment of magic from Salah.
Last season's Premier League Player of the Year cushioned Roberto Firmino's pass in off the far post for his second goal of the campaign.
But Liverpool had to withstand a few nervy moments with �65-million goalkeeper Alisson Becker producing a vital save to deny Pascal Gross two minutes from time to seal a third consecutive clean sheet.
City ensured Emery was handed a baptism of fire at the Emirates a fortnight ago and defeat in a five-goal thriller at Chelsea last weekend put the Spaniard under pressure to finally get points on the board.
His Arsenal side faced West Ham, who are also pointless despite a near �100-million investment in nine new players, and the Hammers went in front thanks to Marco Arnautovic's fine long-range strike.
However, after Nacho Monreal had levelled, Arsenal got the luck Emery needed to strike back as Issa Diop could only turn Alexandre Lacazette's cross into his own net and Danny Welbeck netted in stoppage time to wrap up victory.
"We are happy with the first three points after two defeats, but it's clear we need to improve," said a far from fully satisfied Emery.
"We need to continue working to not concede many chances. We have to see the reasons why we conceded so many."
Richarlison scored three goals in his first two Everton games to dismiss doubts over his price tag. But the Brazilian is now set to miss the next three games after seeing red for a soft headbutt on Charlie Daniels in the first half at Bournemouth.
Despite being a man down, Everton went in front via Theo Walcott before the numbers were evened up by a red card for Adam Smith.
Michael Keane doubled the visitors' advantage but Bournemouth struck back to go level on points with City at the top of the table through Joshua King's penalty and Nathan Ake.
On an afternoon dominated by red cards, Pierre-Emile Hojbjerg's second booking for diving cost Southampton as Leicester fought back to win 2-1 at St. Mary's thanks to Harry Maguire's deflected 92nd-minute winner.
Huddersfield and Cardiff showed why they are favourites to be relegated in a 0-0 devoid of quality with Jonathan Hogg's dismissal the only action of note.