The 40-year-old former Manchester United striker has signed a rolling contract with the south Wales club and succeeds Malky Mackay, who was sacked on December 27 after his relationship with Tan broke down.
Tan sacked Mackay’s head of recruitment, Iain Moody, in October and his decision to part company with Mackay sparked angry protests from some Cardiff fans.
The Malaysian also created controversy in 2012 when he forced Cardiff to change their traditional blue colours to red, but Solskjaer described reports that former United manager Alex Ferguson had warned him not to take the job as “absolute nonsense”.
“He has wished me the best and given me some good advice, as he always does. I had a good conversation with him,” Solskjaer told journalists at his introductory press conference.
“I have had great chats with the owner and (chairman) Mehmet (Dalman) and they have told me they want me to be in charge of the football and for me I do not need any more confirmation,” the Norwegian added.
“I go into this with my eyes open and I want the fans to be proud of us when we go on the pitch.
“I will only ask my players for 100 per cent, which was hammered down to me with the gaffer and Roy Keane at Manchester United. We can forgive mistakes as long as we have a go and enjoy playing in the Premier League.
“I feel lucky to be back in the Premier League. I had to have a talk about it with the family, obviously, but it is a great opportunity.”
Cardiff were promoted to the Premier League after a 51-year absence last season, but Solskjaer arrives with the south Wales club one point and one place above the relegation zone.
He will bring in former United coaches Mark Dempsey and Richard Hartis to work alongside him at the Cardiff City Stadium.
Solskjaer watched Cardiff lose 2-0 at Arsenal on Wednesday and he said he wanted the team to continue to play attacking football.
“I have a positive outlook on football,” he said. “I want the players to express themselves. They love playing football, getting forward, running forward.
“My brand of football is forward-minded and we saw yesterday (at Arsenal), when you secure the ball and give it to Jordon Mutch and have Fraizer Campbell, Kim Bo-kyung and Craig Noone running forward, they cause problems, so we want forward football.”
Solskjaer spent 11 years as a United player after signing from Molde in 1996, during which his boyish looks and razor-sharp instincts in front of goal earned him the nickname ‘The Baby-faced Assassin’.
He was renowned for coming off the bench to score vital goals, notably netting a famous injury-time winner against Bayern Munich in the 1999 Champions League final in Barcelona.
He joined United’s coaching staff after retiring in 2007 and was appointed manager of Molde three years later, leading the club to glory in the Norwegian top flight for the first time in their history in 2011.
They successfully defended their league title the following year and also won the Norwegian Cup in 2013.
Dalman said that Solskjaer was the club’s first-choice target to succeed Mackay, saying: “There was only one name in my head from the very beginning. We’re delighted to welcome Ole.”
Solskjaer looked to endear himself to Cardiff’s fans by identifying success against fierce local rivals Swansea City as one of his top priorities.
“I would like us to finish above Swansea,” he said with a grin. “That must be an aim. That will be important for everyone here.”