In a post on Instagram the Austrian drag queen said she has been living with the virus for many years.
She said she had decided to come out with the news herself before her friend did.
Wurst, who won the singing contest in 2014 with the song Rise Like A Phoenix, wrote that she had received medical treatment and that she was feeling healthy and strong.
The bearded performer, the alter ego of Thomas Neuwirth, wrote that "coming out is better than being outed by a third party".
She said in the post in her native German: "I have been hiv-positive for many years. this is actually irrelevant to the public, but an ex-boyfriend threatens me to go public with this private information, and I will not give anyone the right to frighten me and influence my life in the future."
She added that her reason for going public was "to encourage (others) and make a step against the stigmatisation of people who through their own behaviour or through no fault of their own were infected with HIV."
The post concludes: "to my fans: the information about my hiv status may be new to you - my status is not! I'm well and... I'm stronger, more motivated and liberated than ever. Thank you for your support!"
Ian Green, chief executive of the Terrence Higgins Trust, which campaigns on and provides services relating to HIV, said: "The decision to talk openly about your HIV status should be a personal one and not taken away or ever, ever used as a threat.
"Threatening to reveal someone's HIV status, under any circumstances, is entirely wrong. What other health condition would be used as blackmail against someone? And we know this isn't something which only happens to those in the public eye.
"Despite all the medical advances we've made in the fight against HIV, in treatment and in testing, Conchita's Instagram post highlights how much more we have to do to tackle the abhorrent stigma which still surrounds the virus.
"We applaud Conchita for handling this with such dignity and including in her post that people living with HIV who are on effective treatment, like her, can't pass the virus on."