Police said they were being held on suspicion of child neglect.
The BBC understands the three men were among four team leaders at the Medway Secure Training Centre, in Rochester, who were sacked on Tuesday.
A further three men at the unit, run by security firm G4S - two duty operations managers and a training centre assistant - remain suspended.
A female duty operations manager has been placed on restricted duties, while a male healthcare worker employed by Central and North West London NHS Trust (CNWL) has also been suspended.
A trust spokesman said: "One CNWL staff member contacted us before the programme to say that they had been present at one incident in the film; the footage broadcast confirmed this.
"Whilst we are pleased he came forward, he has been suspended whilst that incident is fully investigated."
The allegations relate to 10 boys aged 14 to 17.
In a statement, G4S managing director for children's services, Paul Cook, said the company fully supported the action of police and was continuing to provide officers and the local authority with full access to the centre and its records, including CCTV footage.
"There is no place in our business for the conduct shown on the BBC's Panorama programme on Monday night.
"We are grateful to the police for their swift action in this case.
"We will work with the police and local authority to keep our own actions under review in light of today's developments," he said.
The arrests follow undercover filming by the Panorama programme at the 76-bed centre, which is for young offenders aged 12 to 17.
The programme highlighted allegations of inappropriate staff conduct.
G4S referred the claims to Medway's local authority designated officer, the Youth Justice Board, and the Ministry of Justice.
Among the allegations uncovered by Panorama and now subject to investigation are that Medway staff:

The 15cm high concrete wall is topped with razor wire, miniature US flags and includes 'Keep Out' signs written in both Spanish and English.
It was created by LA-based street artist Plastic Jesus and was inspired by comments made by the Republican Presidential nominee last year when he outlined plans to build a wall along the US-Mexico border to stop illegal immigrants.
Curious tourists snapped photographs of the Los Angeles structure, while a picture of the unusual artwork posted on the artist's Instagram account received more than 14,000 likes by Thursday morning.
One person called the wall "the most genius vandalism of the star yet," while another said it was the "best thing I've seen all day."
There were also those who were unimpressed by the Trump-attacking artwork, with one user exclaiming that "A Trump presidency is what the US needs."
In an interview with the BBC, Plastic Jesus explained why he had built the 'wall':
"I wanted to create a piece that drew attention to [Trump's] potential policies," he says. "I knew this week was the republican convention and I suppose I was capitalising on the momentum of the convention."
And so, on Tuesday afternoon he took his tiny wall made from wood and concrete, and placed it on top of Donald Trump's star on the Hollywood Walk of Fame.
He says the response to his wall has been largely supportive. "I went to Hollywood Boulevard yesterday to take photographs and found myself in an absolute bottleneck of people taking photographs of the piece - it's been absolutely crazy."
The artist, who emigrated to the US from the UK nine years ago, says he's not a fan of Donald Trump, "In the US you'll interact with immigrants from all over the world and it's one of the things that makes the US so great. Donald Trump's policy proposals are a threat to all of the immigrants."
Donald Trump received the 2,327th star on the Hollywood Walk of Fame in 2007 for his work on America's The Apprentice.
By Daniel Avis, BBC's UGC and Social News team
Three men have been arrested following a BBC Panorama investigation into abuse at a young offenders centre in Kent.
[NEXT_CONCEPT]
Donald Trump's controversial call for a "great, great wall" on the US-Mexico border has yet to be realised but a tiny, tiny one has been built instead - surrounding his star on the Hollywood Walk of Fame.