Solskjaer replaced Malky Mackay as boss last month but a poor run of form - culminating in the 4-0 home defeat to Hull City last weekend - has left Cardiff three points adrift of safety in 19th place in the Premier League table.
Tan said: "Although we're not doing very well now, we are in the relegation zone, I'm convinced and believe that Ole Gunnar Solskjaer will be able to keep us in the Premier League."
The Malaysian businessman also insists that he retains the backing of the majority of Bluebirds fans and believes he has been portrayed unfairly as a villain after sacking Mackay.
"Do you think that Malky Mackay would have got us promoted without my investment? Without me giving him money? After we got promoted the fans were all very supportive, everybody was appreciative," he told BBC Sport.
"Overall, even now, I think, I would say even 90 per cent of the fans are supportive, the problem is you have 10 per cent or five per cent or maybe even a few hundred people that are not supportive, who are more vocal.
"And then you have the British press who have been misled by some people, feeding them wrong information and saying what a terrible person I am and trying to make fun of me and paint me like a villain.
"But my experience is the majority are very supportive. I go to the club, many people come and shake my hand, want to shake hands with me, want my autograph, so generally I think it is not too bad.
"I will say that the British press is unfair and maybe they are unfair because we didn't tell our side of the story that well.
"And sometimes the British press (are) maybe a little bit racist and quite unfair."
Despite the criticism, Tan insists he is in it for the long haul at Cardiff and believes that eventually the fans who are against him will be made to eat their words.
"The club would have gone bust many times if I had not been around," he said. "I came to invest in the club and I just put in some money and I thought that's about it, actually I didn't know the club was in such bad shape.
"I made a decision and when I make a decision to do something I want to stick by it and see it through to the end and make it a success. And likewise the same for Cardiff.
"We were going through some difficulties, we had the wrong people doing things for us but I think we will get it right.
"And I think when this thing is over, fans will understand and they should regret what they have done to me and maybe some of them should apologise to me."
Meanwhile, Cardiff fans have asked Tan to promise he will not follow the example of Hull City owner Assem Allam and consider changing the name of the club.
"We would like a firm commitment from Mr Tan that a Hull City-style change of
name will not be countenanced by him at any time in the future," said Cardiff City Supporters' Trust chairman Tim Hartley.