Cardiff sit six points off safety in 19th, with only five games to play, after Jason Puncheon's brace and a Joe Ledley goal condemned them to a home loss against fellow strugglers Palace.
If Sunderland gain a point at Tottenham on Monday, Cardiff would slip to the bottom of the table and Solskjaer has acknowledged his team face an uphill battle to secure their top-flight status.
"We've got to work hard; we've got to try and turn every stone because we need a miracle. That's what we need to survive," the Norwegian told Sky Sports.
Having picked up a point at West Brom last week in a dramatic 3-3 draw, Cardiff rarely threatened to claim a positive result on Saturday.
A dejected Solskjaer pulled no punches in his assessment of the game.
"It's very disappointing. We worked hard all week. It's very disappointing for the fans and the players. We were really looking forward to the game," he added.
"(Staying up) was (going to be) difficult - that makes it even more difficult, this result, this performance.
"But we've got to be full of fight. It's been a tough season for everyone involved. We were tired towards the end, we ran out of ideas. It's been an up and down season."
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