Manchester City's perfect start to their Premier League title defence was ended with a 1-1 draw at newcomers Wolverhampton Wanderers.
City, who won their first two games of the season in fine style, again largely dominated against Championship winners Wolves at Molineux but still had to come from behind to earn a point against Nuno Espirito Santo's impressive side.
Wolves had gone ahead fortuitously after the break with Willy Boly's diving header actually going in off his arm only for Aymeric Laporte to then equalise with a header, his first goal for City.
The champions hit the woodwork three times, once through Raheem Sterling, whose thunderbolt was tipped brilliantly on to the woodwork by Rui Patricio, and twice from Sergio Aguero, whose free kick struck the crossbar in the dying seconds.
Manager Pep Guardiola felt his side had been let down by fragile defending but reckoned it was still a good point against a team that had demonstrated why they could still thrive in the top flight.
Arsenal prevailed after two defeats under Emery but only after Marko Arnautovic's first-half goal for West Ham prompted brief thoughts that the home side might lose their first three league matches of the season for the first time since 1954-55.
Arsenal, though, responded quickly with a goal from Nacho Monreal, before a second-half own goal from Issa Diop and a Danny Welbeck clincher in injury time left the Hammers still searching for a first point under new boss Manuel Pellegrini.
It was the worst start by a West Ham manager since Avram Grant lost his opening four league games in 2010-11.
Mohamed Salah's 23rd-minute strike was enough to earn Liverpool a 1-0 win over a dogged Brighton and Hove Albion and give them an early top spot in the Premier League with three wins out of three.
It was hardly vintage Liverpool, with Brighton fresh from their victory over Manchester United last week, well-organised at the back and frequently frustrating Juergen Klopp's front three.
But it was the kind of hard-fought three points that Liverpool will need to pick up throughout this season if they are to mount a real challenge to Manchester City in the title race.
Liverpool made a strong start, and Brighton's Australian keeper Mathew Ryan did well to keep out a Roberto Firmino header before Trent Alexander-Arnold rattled the crossbar with a 15th-minute free kick.
The breakthrough came when Milner robbed Yves Bissouma as Brighton tried to play the ball out from the back and a quick exchange of passes from Sadio Mane and Firmino ended with a clinical finish from Salah, last season's top scorer with 32 league goals.
Brighton kept a disciplined formation, frustrating Liverpool but creating little until a Pascal Gross header in the 88th minute forced a good save out of Liverpool keeper Alisson Becker.
Bournemouth and Everton maintained their unbeaten starts to the season in a dramatic match at Dean Court in which both sides ended with 10 men.
Everton's new Brazilian strike hero Richarlison was dismissed for a headbutt aimed at Adam Smith just before the break while Smith himself was sent off for a second half foul on Theo Walcott.
The home side then fought back after Walcott and Michael Keane had put Everton two up, with goals in the last 15 minutes from Josh King, from the penalty spot, and Nathan Ake.
Southampton's difficult start continued with Harry Maguire's late winner seeing Leicester City earn a 2-1 comeback win, a victory that will have brought much pleasure to visiting manager Claude Puel on his return to his former club.
Southampton, who have won just once at home in the Premier League this year, also had midfielder Pierre-Emile Hojbjerg sent off for two yellow card offences.
Huddersfield Town and Cardiff City did nothing to suggest they will be anything but relegation contenders this season as a goalless draw left them still looking for a first win of the season.
Huddersfield did at least earn their first point of the season but only after their captain Jonathan Hogg was sent off for a headbutt aimed at Harry Arter.