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Wenger is out of contract at the end of the season and has been offered a new two-year deal, although he is yet to announce whether he will continue.
Sixth-placed Arsenal are seven points off the top four with eight games left.
"I've managed over 1,100 games for Arsenal and we're not used to losing like that," said the Frenchman.
"We have to respond very quickly and not accept it."
Some travelling fans told Wenger it was "time to go", also singing "you're not fit to wear the shirt" at the players, as Arsenal were convincingly beaten at relegation-threatened Palace.
Goals from Andros Townsend, Yohan Cabaye and Luka Milivojevic inflicted a fourth straight away league defeat on Arsenal - for the first time during Wenger's 21-season reign.
It leaves the Gunners in serious danger of missing out on a top-four finish for the first time under the 67-year-old.
"I understand our fans are disappointed and we all are deeply disappointed," he said.
"It's very worrying and disappointing the way we lost the game. Palace were sharp, they beat Chelsea the other day, and that shows they have quality.
"We are in a difficult position. The game tonight doesn't help."
Arsenal controlled possession against their 16th-placed hosts, having 72% of the ball, but were unable to make that dominance count.
The Gunners managed just three shots on target, all in the first half.
"That's not Arsenal," said stand-in captain Theo Walcott. "It wasn't us at all.
"All we can do is apologise for that performance.
"Palace just wanted it more. You could sense that from the kick-off.
"We thought we had got out of this little patch and hopefully we haven't been dragged straight back into it. Judging on that performance, it looks like we have."
Arsenal have won only one of their past five Premier League matches, losing three and conceding 11 goals in the process.
Palace boss Sam Allardyce said he had targeted their defensive frailties in the build-up to Monday's game, highlighting the space behind their full-backs for Townsend and Wilfried Zaha to exploit.
"We all know they are in a poor spell of results for the first time for years," said the former England manager.
"The weaknesses with Arsenal have been defensively because they leave Shkodran Mustafi and and Gabriel really exposed.
"Nacho Monreal and Hector Bellerin play like right and left wingers, the wingers come inside with the centre-forward and they're just left on their own."
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Former Blackburn and Chelsea striker Chris Sutton on BBC Radio 5 live:
"At one time, Arsene Wenger managed the Invincibles. He is now managing the Invisibles. He has to go, because the players are not listening.
"The biggest problem with Arsenal is that it's Wenger who makes the decision on his future. I don't get it. It should be up to the owners.
"They must be embarrassed by tonight's performance. It was limp. They were played off the park by a team in danger of relegation.
"It was a pathetic showing."

The 29-year-old has featured in 53 U's games since joining in 2015, 18 of which have come this season.
"He came and requested the opportunity to leave for personal reasons," said director of football Tony Humes.
"We have been working with him over the past few weeks to resolve the situation in the best way possible."

Wiltshire Police said it happened just before 1800 BST on Wednesday at the junction with High Street in Codford.
The motorbike was travelling south towards Salisbury when it was in collision with the car, police said.
The motorcycle rider was a 49-year-old man and the car driver was a 61-year-old man. Both were local, and were pronounced dead at the scene.
The road was closed for six hours while police carried out an investigation.
Arsene Wenger says the uncertainty over his future is not affecting his Arsenal players, but admits their 3-0 defeat at Crystal Palace is "a big worry".
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Colchester United midfielder Owen Garvan has had his contract with the League Two club cancelled by mutual consent for personal reasons.
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A motorcyclist and a car driver have been killed in a crash on the A36 near Warminster.