A fourth child only has 'a very low level' of the virus after receiving the same drugs.
Further details of the study are expected to be released when the findings are officially announced at the Aids 2014 conference in Melbourne, Australia, which runs from July 20.
"At the moment, the doctors do not know whether they are in fact cured," said AIDS 2014 co-chairman Professor Sharon Lewin to The Daily Mail Australia. "The only way they can tell is if they stop the anti-HIV drugs and see if it comes back.
"We are excited about this though because all four received very early treatment after delivery, and when doctors tried to locate the virus they could find virtually no viruses in their systems.
"These baby cases, and more cases amongst adults, tell us that if we treat HIV very early in some people were able to stop it."
Professer Lewin added: "However you must keep in mind that the best way to stop babies getting HIV is to treat the mother during pregnancy."
The four Canadian babies involved in the trial were all born to HIV-positive mothers and had the virus present in their systems at birth.
They were given high doses of three antiretroviral drugs within the first few hours after they were born.
Following the treatment the virus could no longer be found in three of the babies, and the fourth only had 'a very low level' of the virus.
It's not yet been revealed how old the babies are now, but they are still receiving the medication. It is not known whether the virus will re-emerge once their treatment is stopped.
The news of the four babies comes just over a year after the first ever case of a baby being 'cured' of HIV was announced in March 2013.