Ms Alqunun went to the Australian embassy in Bangkok on Wednesday and completed the visa application.
But Australian officials confirmed to the ABC that her case was still being processed late on Friday.
There was growing concern within the UNHCR over Ms Alqunun's security as she remained in Bangkok.
With these security fears in mind, she was taken to the Canadian embassy in Bangkok on Friday morning. Her visa was processed within several hours and she was booked on a flight to Canada in the afternoon.
"When referring cases with specific vulnerabilities who need immediate resettlement, we attach great importance to the speed at which countries consider and process cases," a UNHCR spokesperson in Bangkok told AP in an email reply on condition of anonymity.
Canada's ambassador had seen her off at the airport, where Ms Alqunun thanked everyone for helping her. She plans to start learning more English.
Too tired to take questions
Her situation has highlighted the cause of women's rights in Saudi Arabia, where several women fleeing abuse by their families have been caught trying to seek asylum abroad in recent years and returned home.
Ms Freeland said Ms Alqunun preferred not to take questions on Saturday.
"It was a pleasure for me this morning to welcome her to her new home," Ms Freeland said.
"She is obviously very tired after a long journey and she preferred to go and get settled. But it was Rahaf's choice to come out and say hello to Canadians.
"She wanted Canadians to see that she's here, that she's well and that she's very happy to be in her new home."
Ms Freeland said Ms Alqunun commented about the cold weather and she responded that it gets warmer in Canada.
Ms Alqunun flew to Toronto via Seoul, South Korea, according to Chief Surachate.
Ms Alqunun tweeted two pictures from her plane seat — one with what appears to be a glass of wine and her passport and another holding her passport while on the plane with the hashtag "I did it" and the emojis showing plane, hearts and wine glass.
Canada's relationship with Saudi Arabia already strainedCanada's decision to grant her asylum could further upset the country's relations with Saudi Arabia.
In August, Saudi Arabia expelled Canada's ambassador to the kingdom and withdrew its own ambassador after Canada's Foreign Ministry tweeted support for women's right activists who had been arrested.
The Saudis also sold Canadian investments and ordered their citizens studying in Canada to leave.
No country, including the US, spoke out publicly in support of Canada in that spat with the Saudis.
Ms Freeland did not address what Ms Alqunun's case would mean to Saudi relations.
"Canada believes very strongly in standing up for human rights throughout the world. We believe very strongly that woman's rights are human rights," Ms Freeland said.
There was no immediate Saudi Government reaction, nor any mention of her arrival in state media.
Ms Freeland said the UN refugee agency found she was in a dangerous situation in Thailand and that Canada is glad they were able to act quickly to offer her refuge.
Ms Alqunun's father arrived in Bangkok on Tuesday, but his daughter refused to meet with him.
The office of the UN High Commissioner for Refugees welcomed Canada's decision.
"The quick actions over the past week of the Government of Thailand in providing temporary refuge and facilitating refugee status determination by UNHCR, and of the Government of Canada in offering emergency resettlement to Ms Alqunun and arranging her travel were key to the successful resolution of this case," the agency said in a statement.
She barricaded herself in an airport hotel room where her social media campaign got enough public and diplomatic support that Thai officials admitted her temporarily under the protection of UN officials, who granted her refugee status on Wednesday.
Chief Surachate said her father — whose name has not been released — denied physically abusing Ms Alqunun or trying to force her into an arranged marriage, which were among the reasons she gave for her flight.
He said Ms Alqunun's father wanted his daughter back but respected her decision.
"He has 10 children. He said the daughter might feel neglected sometimes," Chief Surachate said.
Phil Robertson, deputy Asia director of Human Rights Watch, cited Ms Alqunun's "courage and perseverance".
"This is so much a victory for everyone who cares about respecting and promoting women's rights, valuing the independence of youth to forge their own way, and demanding governments operate in the light and not darkness," he said in a statement.
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