The breach occurred when OpenAI ran training exercises to rate the performance of AI models.
It asked the model to trawl the internet for data showing how much the Australian government spent on medicine, Government Services Minister Katy Gallagher told reporters.
The San Francisco-based company did not alert the Australian government until this month, when it sent an email to a generic government inbox.
"That email address is looked at once a day," Gallagher said.
"We have someone who goes and has a look through. It sometimes gets a number of notifications; sometimes many of them are hoaxes."
Defence Minister Richard Marles said the AI model had "scaled the fence".
"It asked a question, the information was not given and rather than leaving at that point, it scaled the fence."
OpenAI said it spotted the breach in August while carrying out an "extensive review" of its AI models.
"Our review found no evidence of patient records being accessed. The information accessed included aggregate health statistics and internal file names," the company said in a statement.
It added that it "identified activity involving several Australian government websites and services as our models attempted to look up answers ... our models took actions we did not intend".
The breach is one of several recent incidents in which OpenAI has disclosed hacks or unauthorised activity involving its AI agents well after they occurred.
In some cases this was because the activity was only detected belatedly, and in others because the company initially elected not to disclose it.
A separate high-profile incident, a mid-July intrusion into open-source AI repository Hugging Face, was only detected about a week after it took place, according to timelines released by OpenAI and independent investigators.
This helped ignite a global conversation about the risks posed by increasingly powerful AI models.
Rivals Anthropic, Google's Gemini, and Meta have also disclosed incidents of their agents accessing external systems.
Some of America's top AI executives, including Altman, have called for a slowdown of AI development, citing, among other things, the threat of devastating cyberattacks by out-of-control agents.