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:
Three men have been arrested following a BBC Panorama investigation into abuse at a young offenders centre in Kent.