After Paul Pogba's spot kick miss for United cost them two points against Wolves on Wednesday, Rashford could not repeat his opening game strike from 12 yards against Chelsea as he struck the post when presented with the chance to level the scores for United 20 minutes from time.
Daniel James seemed to have spared his blushes cancelling out Jordan Ayew's first-half strike when he curled an equaliser home a minute from time, but Palace went down the other end and Patrick Van Aanholt scored the winner.
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In a match where his side enjoyed 71% of possession and could only direct three of their 21 shots on target, the Red Devils boss was keen to share the blame for the loss.
"We just were not clinical enough in front of goal and they scored two very easy goals," Solskjaer told the BBC.
"I thought we got our act together second half, we just couldn’t hit the target. We have dominated the game but we have not controlled it.
"We should have dealt with the last five minutes better. We defended poorly at times today and it cost us. Sometimes you do not find that end product and that happened today. We did not test their keeper enough. It cost us dearly today."
Pogba losat the ball to Wilfried Zaha in the lead up to Van Aanholt's goal which saw De Gea only able to parry the fierce striek into his own side-netting.
"I think we played really well," Van Aanholt said. "I think we deserved the three points. We dug in and luck was on our side today.
"I was gutted when that Manchester United goal went in. We had worked so hard to get the clean sheet but I am delighted that we could counter and get them back."