Palace have paid £15m plus £5m in add-ons for the 26-year-old, who was a substitute during United’s win over Nottingham Forest on Saturday.
He was on loan at Forest last season but they signed Matt Turner from Arsenal earlier this month.
Henderson said: “There’s something being built here and I really want to be a part of it.”
Henderson is Palace’s third senior signing of the summer after Colombia midfielder Jefferson Lerma and Brazilian forward Matheus Franca.
“This club has got great fans and has a lot of top players who are hungry to succeed,” he added.
“I’m absolutely delighted and buzzing to be here. I can’t wait to get started.”
Henderson has been with United since the age of 14 but never established himself as the team’s first-choice keeper, going on loan spells to Stockport, Grimsby, Shrewsbury and Sheffield United, as well as Forest.
Another former Manchester United keeper, Sam Johnstone, has started the season in goal for Palace and the 30-year-old was named in the latest England squad on Thursday.
“Dean is a fantastic addition to our squad and goalkeeping unit,” said Palace chairman Steve Parish. “We have been long-term admirers.”
Manchester United signed Andre Onana to replace David de Gea as their first-choice keeper this summer and are set to buy Fenerbahce’s Altay Bayındır for £4.3m to replace Henderson as their back-up option.
“Then we’ll move on to the next game and then with both of them knocking on my door saying, ‘I’m the No1, I’m the best goalkeeper’, I will have to make a decision which one to play.”
Johnstone has established himself as second-choice goalkeeper for England behind Jordan Pickford.
But he and Henderson may be competing for a spot in Gareth Southgate’s squad for Euro 2024.
“Sam came into the team when Vicente Guaita got injured”, said Hodgson. “He kept his place, which relegated Guaita to the second spot which he didn’t accept, so we ended up losing him — unfortunate because he was a good goalkeeper.
“The club found Dean Henderson, and he had to wait. He got injured on [his debut], so that wait was partly in rehabilitation. But he would probably have had to wait even if he hadn’t been injured, because Johnstone had the shirt at the time.
“Now Dean Henderson got into the team again because of Johnstone’s injury.”
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