Norwegian Ole Gunnar Solskjaer . AFP
Eric Bailly's own goal and Bernardo Silva's strike on the stroke of half-time were the difference on the scoreboard.
But there was a chasm between the teams as only a string of saves from David De Gea prevented United from a thrashing similar to the 5-0 humbling by Liverpool less than two weeks ago.
"Some praise has to go to them but we were not at our level. Standards need to be raised," said Solskjaer, whose position as United boss is under intense scrutiny.
The Norwegian insisted he will stay on until told otherwise by the club's hierarchy.
"We need to be on the front foot. I can't look at myself and say this is the way I want Man Utd to play," added Solskjaer.
"I have good quiet communication all the time with the club.
"I want the best for Manchester United. As long as I'm here, I want to do what I can do to improve this."
Solskjaer bought himself some time with a 3-0 win at Tottenham last weekend.
But United have won just one of their last six Premier League games to fall out of contention for the title.
Much more was expected after finishing second to City last season and strengthening in the transfer window with the signings of Cristiano Ronaldo, Jadon Sancho and Raphael Varane.
Had things gone differently in the final days of August, Ronaldo could have been lining up for City as they pursued the 36-year-old to fill the void left by Sergio Aguero.
Ronaldo has scored nine times in 13 appearances since rejoining the club, but United have won just five of those matches as Solskjaer has failed to find the right balance from a star-studded squad.
Despite costing £73 million ($98 million) from Borussia Dortmund, Sancho has been shunned in recent weeks and started once more on the bench, while the injured Varane has been badly missed.
- 'Put the ball in the fridge' -
Bailly was included for the first time in the Premier League this season as Solskjaer tried to patch up a leaky defence.
But it took only seven minutes for the Ivorian to give City the perfect start when he sliced Joao Cancelo's cross into his own goal.
United trailed 4-0 at half-time for the first time in the Premier League in their last home game and history could easily have been repeated but for De Gea's heroics.
The Spaniard spread himself to miraculously block from Gabriel Jesus before saving from Cancelo, Kevin De Bruyne and preventing a second own goal, off the boot of Victor Lindelof.
However, even De Gea was at fault for City's second when Silva stretched a leg to meet Cancelo's cross and the United stopper flapped the ball into his own net.
Boos greeted the half-time whistle as the travelling City support gleefully chanted "Ole's at the wheel" to mock the United boss.
The pattern continued after the break City's passing triangles left United chasing shadows in the Manchester rain, summed up by former United captain Gary Neville as a "calm annihilation."
"We spoke about how United are a transition team," said City boss Pep Guardiola. "We had to put the ball in the fridge. A lot of passes and at the last moment we arrive there."
Phil Foden hit the post and Jesus had strong appeals for a penalty waived away as City could easily have won by far more.
"We had the game of our lives today," said Foden. "Everything went right, a lot of possession and chances. We dominated from start to finish."
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