The Bluebirds have now lost five of their last six winless matches and sit bottom of the Barclays Premier League following the reverse at the Etihad Stadium.
The home side took the lead in the 14th minute when referee Neil Swarbrick - in a demonstration of the effectiveness of the Goal Decision System - indicated that Edin Dzeko's shot had crossed the line before Kevin McNaughton cleared.
Craig Noone scored a brilliant equaliser after a weaving run but Jesus Navas quickly responded.
Cardiff battled early in the second half but Yaya Toure and substitute Sergio Aguero made victory safe in the last 15 minutes before Fraizer Campbell grabbed an injury-time consolation.
Solskjaer said: "We felt that with 15 minutes to go we had the crowd turning - not against them but they weren't sure of the three points.
"We made them make a few mistakes. It was a great response after the poor start we had.
"That start was disappointing, again. That mindset has got to change. We have got to come here and expect to play better than we did."
Cardiff's players complained after the award of the first goal, not about whether the ball had crossed the line but apparently because they felt David Silva had handled in the build-up.
Solskjaer said: "It is hard to say if it was a handball. If the referee makes a decision we can't feel sorry for ourselves. He makes the decision."
The Norwegian, appointed earlier this month, claims he was well aware of the battle he was likely to be facing to avoid relegation.
Solskjaer said: "I knew what I was up against when I came here.
"I didn't come here and think everything was going to be all-singing and all-dancing.
"It is a tough league but it is five points up to number 10, so still as tight as it has been.
"We have just got to make sure we keep performing like we did today."
Cardiff's next league game is against Solskjaer's old club, Manchester United, at Old Trafford.
Solskjaer said: "Every single one is a game you can take something from, as that might be the little point that can take you over the line towards the end.
"We will come back to Manchester in 10 days' time and give it a go."
- Sports & Recreation
- Barclays Premier League
- Neil Swarbrick
- Edin Dzeko
- Jesus Navas
- Sergio Aguero