The decision was made public by Australia's Department of Home Affairs, which said it would consider the referral from the United Nations in the usual way.
However, officials in Australia hinted that her request will be accepted. She told Reuters she was fleeing her family's "physical, emotional and verbal abuse", adding she was restricted from travel and continuing her education.
A spokesman for the U.N. High Commissioner for Refugees at its Geneva headquarters, Babar Baloch, said Tuesday it's premature to say what will happen next, but that it could take several days for the agency to look into Alqunun's claims.
She said she was held after leaving her plane in Bangkok and told she would be sent back to Kuwait.
Who is Rahaf al-Qunun?
Ms Mohammed al-Qunun's twitter account later quoted a Saudi diplomat in Bangkok saying it would have been better to take her phone than her passport.
Australia national broadcaster ABC reported that the country's Home Affairs Department announced late Tuesday that it would consider Alqunun's application for asylum if she was found to be a genuine refugee, and called on the Thai authorities and UNHCR to assess her claim as quickly as possible. She told the world defiantly: "I'm real and exist".
She refused to meet her father and brother, who had travelled to Bangkok, Thai immigration chief General Surachate Hakparn said.
"She is now under the sovereignty of Thailand; no-one and no embassy can force her to go anywhere", he said.
After grabbing worldwide attention with dramatic posts on social media in which she said she feared for her safety if made to return home to her family, Alqunun eventually was placed in the care of the UNHCR as her bid for refugee status was considered.
Ms Alqunun garnered global attention when she took her plight to social media this week, tweeting that she had "nothing to lose".
Alqunun's case has again highlighted the cause of women's rights in Saudi Arabia.
"I want Canada to give me asylum!" she tweeted in the early morning on Tuesday. He described the man as being a governor in Saudi Arabia.
Thailand is not a signatory to a United Nations convention on refugees, and asylum seekers are typically deported or wait years to be resettled in third countries.
Lawmakers and activists in Australia and Britain urged their governments to grant asylum to al-Qunun, who was finally allowed by Thailand to enter the country yesterday. She said family members physically abused her often. "She is 18 years old, she has an Australian visa, and she has the right to travel where she wishes and no government should interfere in that".
"Thailand is concerned about their diplomatic relations, their political relations, their economic relations, and their military relations with certain countries,"says Emily Arnold-Fernández, executive director of Asylum Access".
Global pressure has mounted on Thai authorities to keep Alqunun safe and to ensure she isn't forcibly returned to the Saudi kingdom, which has been subject to worldwide condemnation over the killing of Washington Post writer Jamal Khashoggi at a Saudi consulate in Turkey. It said the embassy is not communicating with the teenager, but is communicating with Thai authorities.
Women trying to escape abuse by their families have been caught trying to seek asylum overseas in recent years and returned home.
The latest incident comes against the backdrop of intense scrutiny on Saudi Arabia over the shocking murder of Saudi journalist Jamal Khashoggi past year, which has renewed criticism of the kingdom's rights record.