The Brighton-based charity sent letters to The X Factor boss asking Cowell and Sony-owed Syco to withdraw its application for ownership over the name.
Syco relented last week, leaving the music charity - which works with vulnerable young people - with £8,000 of legal fees.
The girl group now named Little Mix assumed the name of Rhythmix in the early shows, a name the charity has held since 1999. The charity allows young people across the South East to develop their personal and social skills through engaging creatively through music.
The band's name changed publicly in October but was still among Syco's trademarks, including The X Factor and Got Talent, until the final push from the charity forced Cowell’s hand last week.
With the help of Unity, on a pro-bono basis, the charity has started a campaign to ask X Factor fans to donate to the charity rather than using money to vote on the talent show.
Unity co-founder Nik Done said: ‘When I heard what was happening with this story I was outraged so I got in contact with Mark Davyd [Rhythmix chief executive] to offer our services.
‘We had a meeting in the morning and by the afternoon we were working on this campaign.
‘Our aim is simply to get the public on board, gain support for the charity’s aim to claim back some of the legal costs, and if in the process we raise the profile of the charity that can only be a good thing.’
A Twitter campaign entitled #CowellMustPay, started by the agency, is also helping the cause and has already gathered celebrity backing. Stephen Fry, Edith Bowman and Alistair McGowan are among those who have pledged their support.