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.
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.
(Reporting by Ian Chadband, editing by Pritha Sarkar)
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