Zaha fired home that retaken spot-kick and the former United player impressively wrapped up a 3-1 victory at Old Trafford, where debutant Donny van de Beek had temporarily reduced the deficit.

"Yeah, you can see we're short and they deserved the points," Solskjaer said. "They were sharper than us in the challenges, tackles.

"We started off slowly and that had a knock-on effect for the rest of the game when they got the first goal.

"We had a long season last season and we didn't finish until August 16, and we've not had enough preparation time to be sharper.

"Of course we've worked hard with the ones who have been here, there's been a few players that have been away on internationals.

"But Roy (Hodgson) said they had four friendlies and this is their third competitive game, which is a bit different to our one against Aston Villa last week."

Executive vice-chairman Ed Woodward was at Old Trafford for the defeat on a day when he spoke of the need to be "responsible in our use of resources during this most extraordinarily challenging time".

But the performance only highlighted the need to add to Van de Beek's acquisition before the October 5 deadline, with fan consternation at the lack of signings palpable online in the absence of attending supporters.

"We can play so much better than today," Solskjaer said. "If you look at the players we've got on the pitch and on the bench and not here today, I think we all need to look within first.

"We need to look at ourselves and we know that we have to perform better.

"Then again, we're always, always looking to improve if there's something out there that's available to the right price.

"We always want to strengthen but I think we can all look ourselves in the mirror today instead."

Solskjaer did not believe Palace should have been awarded a penalty for handball against Victor Lindelof but understood why it was retaken, even if the United boss felt that decision was "harsh".

The 1999 treble hero accepted his side were second best on Saturday evening as the Eagles started a top-flight season with back-to-back wins for the first time in their history.

"It was a very difficult start," said Palace boss Hodgson, who also oversaw victory at Old Trafford last season.

"When the fixtures came out, we saw Southampton, who of course ended last season so well and had a brilliant away record, and a trip to Old Trafford, and then Everton, who we knew were strengthening enormously, and then Chelsea away.

"We were looking down the barrel of a gun to some extent, especially with all our injuries in defence.

"But we've survived, managed the first two games extremely well and that's very good because those six points in the bag no-one can take from us."

Hodgson is also hoping that nobody will take Zaha away after the former United attacker made it three goals in two league games.

"I'm very pleased for him and very pleased for the club he plays for," he said.

"I don't think one should always link performances to what has gone on in someone's life six years ago. As far as I'm concerned it's about the present and about the future.

"I'm delighted with Wilf's present and I'm looking forward hopefully to working with him in the future and seeing more performances like today."