The boys, who have not been named, say the woman encouraged them to film her having sex with each of them, according to Eskilstuna Kuriren.
She then urged them to watch the films and to phone her when they missed her, they said.
The Swedish newspaper was shown the films and believes they show the acts described by the boys, and that the woman can clearly be identified, although she rejects all the accusations.
The pair said they tried to tell social services and the police but nobody listened to them.
They then went to the newspaper with their story on the advice of relatives living in another part of Sweden.
'The boss at the home forced us into it and exploited us for sex. She knew we had to, and that nobody would help us,' one said.
The boys claim they had sex with the woman on four or five occasions, at a hotel and in her own home in the Sörmland region.
On one occasion all three were in bed together, they say, but mostly one of the boys had sex with the woman while the other filmed.
The woman also offered them alcohol, they claim, saying it would help them enjoy the experience, despite them being under the legal drinking age of 18.
They allege that she promised to buy them clothes and toiletries if they had sex with her, but she also told them she would destroy important documents and get the migration agency to deport them if they said 'no'.
They also faced sexual advances from other people in the woman’s social circle, they say.
One of the boys said they would be 'stoned to death' if they ever went back to Afghanistan and their story got out.
The boys were recently moved to a home in a different municipality where local authorities contacted the police after hearing their allegations.
Police confirmed to the newspaper that an investigation into sexual offences was ongoing, although they would not specify what the alleged offences were.
The boys say they arrived in Sweden in November after they were urged by their families to flee Afghanistan and seek a better life.
Two Afghan migrant boys say Swedish woman who ran refugee home forced them to have sex with her on camera or she would have them deported
Boys claim they had sex with the woman on four or five occasions
One said they would be 'stoned to death' if they went back to Afghanistan
They tried to tell social services and the police but nobody listened
By MAILONLINE REPORTER
PUBLISHED: 09:28 EST, 16 June 2016 | UPDATED: 12:54 EST, 16 June 2016