The Federal Government has been ordered to pay a Tamil family more than $200,000 in legal fees over their bid to stay in Australia.

Priya and Nades Murugappan and their Australian-born daughters Kopika and Tharunicaa, aged four and two, have spent more than two years in detention fighting their deportation to Sri Lanka.

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Federal Court Justice Mark Moshinsky ordered the family's costs of $206,934.33 be paid after they successfully argued their youngest daughter was denied procedural fairness in an application for a protection visa.

Immigration Minister David Coleman lifted a barrier to allow him to consider a visa application for Tharunicaa last May, but no decision was made.

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He asked for a briefing on the family's case, including his ability to exercise a power allowing her to apply for a protection visa.

But an assessment by an immigration department official in August found it was unlikely Australia's protection obligations would apply to the family.

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"The applicant was not notified that the August 2019 assessment was being conducted and was not invited to comment in relation to any aspect of the assessment," Justice Moshinsky said.

Priya and Nades fear persecution if they are returned to Sri Lanka.

The family had been held in detention in Melbourne but were transferred to Christmas Island in August after an urgent court order put a hold on their deportation.