Canada also sparked fury in Riyadh by demanding the "immediate release" of jailed rights campaigners, including Samar Badawi, the sister of jailed blogger Raif Badawi, whose family lives in Quebec.
Home Affairs Minister Peter Dutton also said there would be no "special treatment" in the case. Human rights activists say many such cases have gone unreported.
After the arrests, Canadian Foreign Minister Chrystia Freeland, tweeted "Canada stands together with the Badawi family in this hard time, and we continue to strongly call for the release of both Raif and Samar Badawi". Australia was among the countries considering her case.
Her father and brother arrived in Bangkok on Tuesday with the intention of seeing her - but Thai police said she refused to meet them.
Following a tense 48-hour stand-off at Bangkok airport, some of it barricaded in a transit lounge hotel room, she was allowed to enter the country and has been processed as a refugee by the United Nations Refugee Agency (UNHCR). Several Saudis girls and women fleeing abuse by their families have been caught trying to seek asylum overseas in recent years, and returned home.
"Yes, Australian has granted her asylum, but we are waiting to hear where exactly she is going", immigration police chief Surachate Hakparn confirmed to CNN.
Ms Qunun alleged that she was abused by her family - who deny the allegations - and rights groups also said she had renounced Islam, risking prosecution in Saudi Arabia.
"If she is found to be a refugee, then we will give very, very, very serious consideration to a humanitarian visa", Health Minister Greg Hunt told the Australian Broadcasting Corp. before the UNHCR's referral.
'Rahaf received death threats and for this reason she closed her Twitter account, please save Rahaf life, ' @nourahfa313 wrote.
"When the United Nations made a request of us that we grant miss (Rahaf Mohammed) al-Qunun's asylum, we accepted".
"He has denied all wrongdoing as accused by the Bahrain government", Nadthasiri Bergman, Mr.al-Araibi's lawyer in Thailand told Reuters.
Australia's Foreign Minister Marise Payne met with senior Thai officials in Bangkok on Thursday.
Immigration chief Maj-Gen Surachate Hakparn address media at Suvarnabhumi Airport where Rahaf Mohammed al-Qunun has barricaded herself inside a hotel room in Bangkok, Jan. 7, 2019. Describing her case as a "family affair", it said Saudi officials had neither seized her passport - as Alqunun had claimed several times - nor demanded her deportation back home.
"Ms.al-Qunun's plight has captured the world's attention over the past few days, providing a glimpse into the precarious situation of millions of refugees worldwide", UN High Commissioner for Refugees Filippo Grandi said in a statement.
Canada's decision to grant her asylum could further upset the country's relations with Saudi Arabia. Al-Araibi, however, faces a court trial in Thailand for extradition as requested by Bahrain for his role in anti-government protests.
"When she arrived, she open a new (Twitter) account and her followers grew to 45,000 in one day", he said in Arabic.
"Canada has granted her asylum".
In 2014 Bahrain sentenced him in absentia to 10 years for allegedly vandalizing a police station.
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