Cardiff dropped a place to 19th following their crushing 3-0 home defeat against Crystal Palace, and they are now six points adrift of possible safety with just five games left.
Former Cardiff player Joe Ledley scored Palace's second goal, in between strikes from Jason Puncheon, as the Londoners continued their spectacular revival under manager Tony Pulis.
"It was a big game for us, and I am very disappointed for the fans and players that we ended up with a performance like this," Cardiff manager Solskjaer said.
"If you produce a miracle - that sometimes happens in football - you are still capable of getting out of this and staying in the Barclays Premier League.
"I am full of fight. It has been a tough season for everyone involved with the club. Today in the second half, it showed a little bit that it has taken its toll.
"There was a lack of belief a little bit towards the end. We ran out of ideas.
"Players are desperate to perform. I have seen them all week, and you are so disappointed for them and the fans that we produced a performance like we did. Of course they want to stay in the league."
It was Palace's second successive victory, arriving just seven days after they toppled title-chasing Chelsea.
Palace, red-hot favourites for the drop earlier this season, are now well clear of trouble. Cardiff, though, can only dream of such heights after they failed to deliver when it really mattered.
They remain firmly entrenched among the bottom three ahead of next Saturday's trip to Southampton, and there can be no excuses following an error-strewn and largely passionless effort.
Quite what watching Cardiff owner Vincent Tan made of it all is anyone's guess, with his team's brief stay in the top flight looking set to end at some point during the next five weeks.
However, Palace remain upwardly mobile under Welsh-born Pulis, who can surely start planning for next season safe in the knowledge that relegation is no longer an issue.
A frustrating afternoon for Cardiff was underlined through their Manchester United loan signing and former Palace favourite Wilfried Zaha being booed when Solskjaer took him off early in the second half.
"You never want to take players off," Solskjaer said. "I felt for Wilf today, especially first half, he had some chances and he was really positive.
"Wilf is a strong character. It must have been a difficult game for him, of course. It was his old club that he played against but he was professional, he did his best and I cannot fault him for his effort at all.
"I have to say Palace are a fantastically well-organised team and hard to break down, which you saw last week and today, and we just did not have the quality and ideas to break them down.
"Today was very hard. We've got to produce one (a miracle) to stay up. I am full of fight. I have been here three months, I have got more energy and fight in me than ever.
"It was a disappointing performance. Things like this give me more fight. Things will have to happen, things will have to change.
"They are professional footballers and they have got to realise they have still got something to fight for. If you do produce a miracle against Southampton, suddenly you might be back here in with a chance.
"Today it will be dark and gloomy, but football changes."
There are no such troubles for Pulis though, who saluted his players.
"It was a good performance," he said. "We knew Cardiff would be strong and get lots of balls in their air, but our two centre-halves and two central midfielders were outstanding and we fancied ourselves on the break.
"We played really well against Chelsea last Saturday and we didn't want to come here after the Lord Mayor's Show.
"It is very rewarding for everyone concerned. That said, we still have six games to go and we are not safe. The game against Aston Villa on Saturday is massive."