Substitute Phil Foden rescued a 1-1 draw at the London Stadium but the disappointing result means Pep Guardiola's team have taken just five points from their past four matches.
City remain favourites to snatch the Premier League title back from Liverpool but there are growing doubts over their ability to establish momentum in a topsy-turvy season for the 'big six'.
Injury-hit City, without Fernandinho, Aymeric Laporte, Nathan Ake and Gabriel Jesus, dominated possession but rarely found top gear against West Ham in the early kick-off.
They are already five points behind leaders Everton, who play Southampton on Sunday, and look more vulnerable than at any time since Guardiola's first season in charge.
West Ham went ahead against the run of play in the first half through Michail Antonio's superb overhead kick from Vladimir Coufal's cross.
Foden, who came on at half-time, restored parity minutes later, coolly controlling Joao Cancelo's cross before slotting home.
But the visitors could not turn their dominance into goals, with West Ham indebted to Lukasz Fabianski, who produced some vital late saves.
"We created enough," a frustrated Guardiola told BT Sport. "Their defenders were defending so deep so it's difficult to find spaces, but we had the chances.
"I'm not a guy who can predict the future. We've already dropped seven points, which is a lot. But we've struggled a lot for many reasons. We have to take things game by game and see what happens."
There could be more grim injury news for Guardiola, who said Argentine forward Sergio Aguero, who has only just returned after a long lay-off, looked to have picked up a suspected hamstring injury.
West Ham manager David Moyes was delighted his side had hung on for a draw after their spectacular comeback against Tottenham last week, where three late goals secured a 3-3 draw.
"I take a fantastic point against a really good team," he said. "The character the players showed, their attitude to stick at it. Manchester City have come here two or three times recently, winning by four or five, so today's performance was much improved."
United, Chelsea challengers?
Manchester United and Chelsea, who both finished in the top four last season, are hoping to take advantage of sluggish starts by City and Liverpool but they are also struggling to find their mojo.
Chelsea's £220 million ($287 million) splurge in the transfer market has so far only produced two wins from their opening five league matches, with United suffering two defeats in their four matches so far.
United, who named new signing Edinson Cavani on the bench at Old Trafford, will be buoyant after bouncing back from their humiliating 6-1 defeat against Tottenham to win at Newcastle and Paris Saint-Germain.
That eye-catching win in Paris was their 10th consecutive away win -- a club record -- but their home form in the league makes grim reading, with an aggregate scoreline of 9-2 against them in defeats by Crystal Palace and Spurs.
In the second match of the day, goals from Jairo Riedewald and Wilfried Zaha for Crystal Palace condemned promoted Fulham to their fifth defeat in six matches.
Tom Cairney pulled a goal back for 10-man Fulham in the final seconds but Roy Hodgson's side held on to win 2-1 and climb to fifth in the table, ahead of the evening games.
Champions Liverpool face Sheffield United later on Saturday in their first Premier League game without Virgil van Dijk, who suffered a knee injury against Everton last week that could rule him out for the rest of the season.
Jurgen Klopp's side, who won 1-0 away to Ajax in midweek, began their title defence in style but have dropped five points in their past two games.