"Yes, the Australian has granted her asylum, but we are waiting to find out where she's going," confirmed immigration chief Surachate Hakparn to CNN.
Hakparn said Canada has also offered Qunun asylum and is waiting for its decision.
The UN refugee agency UNHCR forwarded Qunun's protection request to Australia on Wednesday, although it had not previously been confirmed in her case.
Hakparn said that Qunun, who is staying in an unknown location in Bangkok, would leave Thailand "almost as soon as the final decision is made."
"We give her the necessary security," he said.
The Australian Interior Ministry declined to comment on CNN, which has also turned to the Canadian government for approval.
. Qunun had flown from Kuwait to Thailand to avoid her family She was afraid that she would kill her because she had given up Islam. She wanted to fly to Australia, but barricaded herself on Sunday in a hotel room at the main airport in Bangkok, after Thai immigration officials tried to push her back to the Middle East.
Qunun and her supporters made an assassination attempt on her social media campaign mostly on Twitter. She documented her arrival and subsequent detention in Bangkok on her smartphone and created new Twitter and Periscope accounts, receiving a flood of supporting messages.
Her story has also put the guardianship laws in Saudi Arabia, which restrict many aspects of women's lives, under international control.
In response to the media campaign, Thai authorities granted access to UNHCR and did not send them to Kuwait. Their online campaign was so successful that Saudi Arabian chargé d'affaires Abdalelah Mohammed A. al-Shuaibi told a Thai translator by Thai officials, "We wish they had seized their phone instead of their passport." Qunun later tweeted the video of the meeting and wrote that her "Twitter account changed the game for what he wanted me to do."
On Friday, Qunun's Twitter account appeared to have been deleted.
Resettlement is not an option for many refugees
The fall of Qunun is unusual due to the rapid offer of resettlement. It is not an automatic right for refugees and, according to UNHCR, less than 1% of registered refugees are relocated every year.
"No one wants to see a young girl in distress, and she has obviously found a safe haven in Thailand," Dutton told reporters in Brisbane.
"We are in close contact with partners about their situation," Maron said. "Canada will always fight for human rights, including women's rights."
Angus Watson of CNN reported from Hong Kong.