The former Arnhem Land schoolteacher served in the Senate for Labor between 1998 and 2013.
Mr Albanese expressed his condolences to everyone mourning the passing of a woman who dedicated herself to bringing new opportunities to the people of the Northern Territory, as a teacher then a senator."As the first woman to represent the NT in parliament, Trish was a trailblazer. But what drove her was making that trail easier for future generations to follow," he said in a statement.
"Trish served her community, our party and the parliament with strength, decency and pride."
Ms Crossin died after having a heart attack on a flight home from the US, where she had celebrated her 70th birthday.
Foreign Minister Penny Wong said Ms Crossin was a resolute advocate for First Nations communities and regional Australia.
"She was also a fierce unionist and feminist, and a co-sponsor of the first marriage equality bill in the Senate," Ms Wong said in a statement.
Ms Crossin fought against the closing down of First Nations bilingual schools in the NT and did early work around compensation for members of the Stolen Generation.
Indigenous Affairs Minister Malarndirri McCarthy, an NT senator, said Ms Crossin always made herself available with wise counsel and words of support.
"She spoke of her personal commitment to reconciliation and she lived that during her time as a senator," she said in a statement.
Ms Crossin's senate career ended in 2013 when she was dropped by then prime minister Julia Gillard in favour of former athlete Nova Peris as the lead candidate for the NT seat.
She is survived by husband Mark, daughters Melinda, Amanda and Kate and son Paul.
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