The 23-year-old Norwegian, who played under Solskjaer at Molde and was sold by the former Manchester United striker to the Dutch club, has joined Cardiff for an undisclosed fee believe to be around £2 million.
"I've known Magnus for many years, having worked with him at Manchester United and then signing him for Molde," Solskjaer told the club website.
"While I was there I had to sell him as he was our best player - and now I feel lucky to get to sign him again for Cardiff City for a reasonable price. I see this as money well spent," he added.
"He's a playmaker, a quarterback if you will, someone who likes to get on the ball and can see a pass. He's technically very good, including set pieces and has great vision.
"Magnus will add a lot to Cardiff City and will blend in very well with the midfielders in the squad."
Eikrem, who never made a first-team appearance at Manchester United after playing for the youth team, has joined Cardiff in time for the league match at home to fellow strugglers West Ham United on Saturday.
City are one place above the drop zone with 18 points from 18 matches, three points in front of the Londoners who are second-bottom.
Solskjaer took over as Cardiff manager at the start of the month as replacement for sacked Malky Mackay who led them up to the top flight last term after an absence of 51 years.
Mackay departed after falling out with billionaire Malaysian owner Tan in a series of public squabbles and despite being acclaimed by fans as the club's best ever manager.