Veterans with amputation-related medical issues can now attend a clinic at the Surrey-based unit's Defence Medical Rehabilitation Centre (DMRC).
The first six such patients have already received treatment there.
Mr Fallon said the veterans who would be treated "have made enormous sacrifices to keep Britain safe".
He added: "This is about making sure they get the best possible treatment."
The six who have already received treatment were seen after referrals from their current NHS centre.
They had all experienced complex amputation-related complications.
While veterans' healthcare remains the responsibility of the NHS, they now have access to the DMRC at Headley Court near Epsom.
Staff there have acquired experience over the last decade in managing the rehabilitation of military casualties with complex trauma injuries.
The new set-up mirrors the tertiary referral system within the NHS, where one hospital can refer a patient to another hospital when needed.
The new programme will also develop existing collaboration with the NHS to ensure that knowledge and best practice are shared.
Veterans will now be treated at Headley Court military rehabilitation unit as well as serving personnel, Defence Secretary Michael Fallon says.