"This is the first time someone living with HIV has been allowed to donate a kidney, ever, in the world," Dr Dorry Segev said in a release.
It was previously thought that HIV carried too great a risk factor for kidney disease in the donor.
But new types of anti-retroviral drugs used to treat the disease are seen as safe for the kidney.
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The patients were "incredibly grateful for this gift and now we just monitor for the long term outcomes," Dr Durand said.
The operation was performed on Monday. Donor Nina Martinez, 35, from Atlanta, told reporters she was "feeling good".
She was inspired to donate her kidney by an episode of "Grey's Anatomy," she said, adding that she was excited to be part of a medical first.
"I knew that I was the one that they had been waiting for," she said. "For anyone considering embarking on this journey, it's doable.
"I've just showed you how and I'm very excited to see who the first follow-on might be."
The recipient chose to remain anonymous but was doing "beautifully", Dr Durand said.
The breakthrough followed another significant development in HIV treatment. In only the second case of its kind, a UK patient's HIV became "undetectable" after a stem cell transplant earlier this month.
There were about 37 million people living with HIV/AIDS in 2017, and the World Health Organization still ranks HIV as "one of the world's most serious public health challenges".