Struggling Burnley struck late to draw 1-1 at Chelsea while Eddie Howe watched a 1-1 draw from the stands in Brighton as Newcastle edged closer to appointing a new boss.
Two weeks after their embarrassing 5-0 humiliation at the hands of Liverpool, United were again palpably second best in a limp display at Old Trafford.
Eric Bailly’s own goal and Bernardo Silva’s strike on the stroke of half-time proved the difference but goalkeeper David de Gea made a string of saves to prevent another drubbing.
United have won just one of their past six Premier League games, giving Solskjaer - and club bosses - plenty to ponder during the international break.
Last week’s 3-0 win at Tottenham gave Solskjaer hope his team had turned the corner and a late Cristiano Ronaldo goal earned them a valuable Champions League point against Atalanta in midweek.
The Norwegian admitted his men, who finished second behind City in the Premier League table last season, were well below par.
But despite the string of poor results, he said he did not feel he was on borrowed time at the club, trophyless since 2017.
“I have good communication all the time with the club and that’s very up front and honest about the situation,” Solskjaer told Sky Sports.
“I work for Man United and I want the best for Man United and of course as long as I’m here I want to do what I can to improve this.”
City deservedly took the lead in the seventh minute when Bailly sliced Joao Cancelo’s cross into his own net.
De Gea kept the marauding City attackers at bay but was at fault for their second when Silva stretched a leg to meet Cancelo’s cross and the United stopper allowed the ball to squirm into his own net.
City boss Pep Guardiola praised his team after the win, which lifted them above Liverpool to second in the table and to within three points of Chelsea.
“The best way to silence Old Trafford is to have the ball with a lot of passes and attack the box in the right moments and we did that,” he said. “We played a really good game.”
Norwich sacked Farke just hours after they beat Brentford 2-1 to register their first three points of the season following their return to the Premier League.
“We feel that now is the right time for a change to give ourselves the best opportunity of retaining our Premier League status,” sporting director Stuart Webber said in a statement.
Farke - the fourth top-flight managerial casualty this season - gave no indication of his fate when speaking after yesterday’s win, hailing the “spirit and belief in this dressing room”.
On the pitch, Chelsea were frustrated as Matej Vydra’s late equaliser ended the European champions’ four-match winning run despite 25 shots from the home side.
The Blues led through Kai Havertz’s first-half header and were 11 minutes away from moving five points clear at the top before Vydra stabbed home from close range.
“Of course I’m disappointed,” said Chelsea boss Thomas Tuchel. “If we played this game 100 times, we’d win 99 times. Today we didn’t win it.
“The second goal would have decided it. It’s always possible you miss one ball, one cross, and this can happen.”
Newcastle, who edged back above Norwich after their draw at Brighton in the late kick-off, are understood to have agreed a deal in principle with former Bournemouth manager Howe to take over at St James’ Park, more than two weeks after Steve Bruce’s exit following a Saudi-led takeover.
But despite the enormous resources at his disposal, Howe faces a huge task just to keep the northeast club in the English top flight.
Crystal Palace are up to ninth after Wilfried Zaha and Conor Gallagher scored in a 2-0 win over Wolves.