The former Manchester United striker, who won two Norwegian Tippeligaen titles in his native country with Molde, succeeds axed manager Malky Mackay with the south Wales club.
Solskjaer takes over at the Welsh club from Malky Mackay, who was sacked by controversial owner Vincent Tan last month following weeks of speculation.
David Kerslake has taken charge of Cardiff's two Premier League games since Mackay's dismissal, guiding the team to a 2-2 draw with Sunderland and a 2-0 defeat at Arsenal.
However, former Manchester United striker Solskjaer will now take the reins for Cardiff's FA Cup tie at Newcastle United on Saturday.
"It's a fantastic challenge for me," Solskjaer told the club's official website. "Cardiff are ready to take the next step up, I hope I can help them.
"I've always dreamed of being a manager in the Premier League and I'm delighted to get the opportunity."
Chairman Mehmet Dalman added: "There was only one name in my head from the very beginning. We're delighted to welcome Ole to Cardiff."
The 40-year-old made his first steps into management at Norwegian side Molde, taking over in 2011 and leading the team to two Norwegian Tippeligaen titles and a Norwegian Cup crown.
He will now attempt to continue that success with the Welsh capital club by ensuring it retains its Premier League status.
Although a key factor in Mackay's dismissal was Tan's dismay at his preseason spending, Solskjaer will reportedly have considerable funds to deal with in the January window.