Solskjaer's men needed a win at St James' Park on Saturday to realistically have any chance of avoiding an immediate return to the Championship.
However, the Welsh side meekly surrendered to a 3-0 reverse, with goals from Shola Ameobi, Loic Remy and Steven Taylor giving Newcastle the points.
Cardiff did have chances to level at 1-0 down, with Newcastle goalkeeper Tim Krul producing a fine reaction save from Kenwyne Jones before Fabricio Coloccini brilliantly cleared an Aron Gunnarson effort off the line.
And Solskjaer was quick to express his disappointment with his side's attacking display after succumbing to the drop.
"(The game) could have gone the other way. We created an unbelievable amount of big chances, but we just didn't take them," Solskjaer told Sky Sports.
"That's the thing in football - you have to take them. First half we should have had three or four (goals). In the second half we should have scored another three or four, at least.
"We've had many injured players, ill players, this season. At some time it's going to catch up."
The Norwegian also praised Cardiff's travelling support and insisted that Cardiff will be going all out to win in their final game of the season against Chelsea.
"We heard (the fans)," Solskjaer added. "Our jobs make such a difference to people's lives - if we could have given them something for the journey back. They're unbelievable. We'll try next week.
"I'm the manager of Cardiff City Football Club. I'm not going to lie down."
