Cardiff's relegation was confirmed after their 3-0 defeat at Newcastle United which, coupled with Sunderland's win at Manchester United, consigned them to a bottom-three finish.
Solskjaer was appointed Cardiff boss in January as successor to Malky Mackay and, despite failing to guide them to safety, he insists he is at the club for the long haul.
Asked if he would be staying, he told Sky Sports: "Definitely.
"I am the manager of Cardiff City football club and I am not going to lay down and feel sorry for myself.
"We will go into the next game showing professional pride and put in a good performance and then we will plan for next season."
Despite the scoreline, Cardiff had a number of chances at Newcastle and Solskjaer feels they should and could have won.
"It should have been the other way, that is just the story of today and we created an unbelievable amount of good chances but we didn't take them, but that is the most important thing in football scoring goals," he continued.
"First-half we should and could have had three or four and at this level you have to take chances and second-half we could have had three or four again and if you add that up that is a lot."
Before the match Cardiff suffered the loss of Mats Moller Daehli, Gary Medel and Kevin Theophile-Catherine through illness - but the Cardiff boss refused to use that as an excuse.
"We have not had many injured, ill players, and sometimes it catches up with you," he admitted.
"It was an important game and we performed well enough but we just didn't take the chances."
He was also full of praise for the fans and insisted they went into the game determined to give them something to shout about.
"That was a lot of the team talk, we can make such a difference to their lives and they have such a long trip and they are unbelievable and we will give them a good game next week and bounce back up next season."