City, who won the first two games of the season in fine style, again largely dominated championship winners Wolves at Molineux, but still had to come from behind to earn a point against the impressive side of Nuno Espirito Santo.
After the break, Wolves had succeeded for the second time with the diving cup of Willy Boly, who actually only went for Aymeric Laporte to equalize with a header, his first goal for City.
The champions hit the woodwork three times, once by Raheem Sterling, whose thunderbolt was brilliantly tipped by Rui Patricio on the woodwork and twice by Sergio Aguero, whose free kick hit the bar in the last seconds.
Manager Pep Guardiola felt that his side was disappointed by fragile defense, but felt it was still a good point against a team that had shown why they could still thrive in the top flight.
Arsenal reigned after two defeats under Emery, but only after Marko Arnautovic's first half of the goal for West Ham had aroused some thoughts that the home team would lose their first three league games of the season for the first time since 1954-55.
However, Arsenal reacted quickly with a goal from Nacho Monreal, before a second-half own goal by Issa Diop and a Danny Welbeck clincher in injury time the Hammers still looking for a first point under new boss Manuel Pellegrini.
It was the worst start for a West Ham manager since Avram Grant lost its first four league matches in 2010-11.
Bournemouth and Everton maintained their unbeaten starts of the season in a dramatic game at Dean Court, in which both sides ended with 10 men.
Evergart's new Brazilian strike hero Richarlison was fired for a headbutt aimed at Adam Smith just before the break, while Smith himself was sent out for a second halffout on Theo Walcott.
The home team then fought back after Walcott and Michael Keane had set up Everton with goals in the last 15 minutes of Josh King, from the penalty spot, and Nathan Ake.
The difficult start of Southampton continued with the late winner of Harry Maguire, who gave Leicester City a 2-1 comeback victory, a victory that visiting manager Claude Puel will have brought a lot of joy upon his return to his former club.
Southampton, who only won home in the Premier League this year, also allowed midfielder Pierre-Emile Hojbjerg to clear two yellow cards.
Huddersfield Town and Cardiff City did nothing to suggest that they will be anything but relegation candidates this season as a goalless draw and that they are still looking for a first victory of the season.
Huddersfield earned at least the first point of the season, but only after their captain Jonathan Hogg was sent for a headbutt aimed at Harry Arter.
(Reporting by Ian Chadband, editing by Pritha Sarkar)