Manchester City offered hope to the chasing pack that the Premier League champions won't be the same unstoppable force as last season in a 1-1 draw at Wolves on Saturday, while Arsenal ended their wait for a victory under new boss Unai Emery.
Everton's flying start to their new era under Marco Silva also 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 City means Liverpool can move two points ahead of their title rivals with victory at home to Brighton later on Saturday.
Wolves opened the scoring in highly controversial circumstances when Willy Boly turned 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."
- Arsenal off the mark -
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 a 3-1 win.
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 Josh 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 came from behind 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 and only noteworthy for Jonathan Hogg's dismissal.