The Sexual Assault Referral Centre (SARC) can gather forensic evidence, even if the person who has been assaulted does not want police involvement.
The service will also provide aftercare and support following an assault, whether recent or historical.
The centre is funded by the West Yorkshire police and crime commissioner and NHS England.
A government report in 2010 called for the setting up of an SARC in every police force area by 2011.
People can be referred to the centre by a number of agencies, including the police. Its location is not being disclosed to the general public.
West Yorkshire Police and Crime Commissioner Mark Burns-Williamson, who represents Labour, said the centre was a "significant step forward".
He said: "It will work alongside the new Independent Sexual Violence Adviser service I have recently commissioned which will provide a much more consistent community-based service, making it easier to report such crimes and receive the aftercare support needed to cope with, and recover from, the consequences of such awful crimes."
A centre to help the victims of sexual attacks in West Yorkshire has opened.