The former Manchester United striker leaves the club after nine months in charge.
Solskjaer was appointed manager in January with the aim of keeping the side in the Premier League but the club were ultimately relegated, and after a poor start to the Championship season - with the club collecting only eight points from their opening seven games.
In a statement, Tan said: "After the Club was relegated many people advised me to let him go, but I decided to keep Ole on for the Championship season. Regrettably our recent results do not justify Ole’s continued role as manager at Cardiff.
“Ole has therefore decided to step down and I have accepted this. During my short association with Ole, I have found him to be an honest and hardworking professional, but unfortunately the football results were not in his favour. As long as I am the owner of this Club, Ole and his family will always be welcome in the Chairman’s Suite as my guests. I wish Ole the very best in his future career in football."
Solskjaer said he was grateful to Tan for giving him the opportunity to manage the club and a difference in philosophy made him stand down.
He said: “I want to thank Vincent for giving me the opportunity to manage Cardiff City FC. He has my full respect and I really wish him all the best in his pursuit of making this football club successful. However, our difference in philosophy on how to manage the club made me decide to step aside and allow the Club to move forward in the direction Vincent wants."