Daehli and Eikrem are former Manchester United youngsters who worked under Solskjaer when he was reserve team manager at Old Trafford and during his reign as Molde manager.
He said Daehli had turned down the chance to return to United, who had first refusal on their former academy player.
"I am pleased to announce we have signed Mats from Molde, we had some competition and I'm glad to say he has chosen to come to Cardiff instead of going back to where he was before," Solskjaer was quoted as saying on the skysports.com website.
"He had talks with United, they had the option to bring him back as part of the deal with Molde, but he has decided to come here and that shows his desire to do well here."
Daehli said the prospect of working with Solskjaer again influenced his decision to join Cardiff.
"Of course it was difficult to say no to Manchester United," he said on the skysports.com website. "I had good years there but I wanted to play for this manager. He said he wanted to take me here. I am delighted to sign and I want to help the team."
Cardiff have dropped to 18th place after Saturday's 2-0 loss to fellow strugglers West Ham United in Solskjaer's first home match in charge since replacing the sacked Malky Mackay, who took them into the top flight last term after a 51-year absence.
(Writing by Sam Holden in London; editing by Josh Reich)
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