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 them to a 2-2 draw with Sunderland and a 2-0 defeat at Arsenal.
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"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 dreamt 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 them to two Norwegian Tippeligaen titles and a Norwegian Cup crown.
And he will now attempt to continue that success with the Welsh capital club by ensuring they retain their Premier League status.
Although a key factor in Mackay's dismissal was Tan's dismay at his pre-season spending, Solskjaer will reportedly have considerable funds to deal with in the January window.