Ms Burney will become the first Indigenous woman to be sworn into federal cabinet on Wednesday.
In delivering the Lowitja O'Donoghue Oration on Tuesday, Ms Burney called on the new opposition leader to help unite and "inspire our country to something really great" by supporting a Voice to Parliament.
"There is no one for whom supporting a referendum for a Voice to Parliament represents a bigger political opportunity than for Peter Dutton," she said.
"An opportunity ... to show his much-talked-about different side.
"It's also about being on the right side of history. And Peter Dutton has in recent days reflected on what it is like to be on the wrong side of history, after walking out on the apology to the Stolen Generations."
Mr Dutton walked out on former prime minister Kevin Rudd's formal apology delivered in parliament in 2008.
Ms Burney said she would be "very pleased" to walk the path of reconciliation with Mr Dutton, and said the Nationals and Greens had the opportunity to place the "national interest ahead of narrow political ambition".
The incoming minister paid tribute to Lowitja O'Donoghue, and credited her leadership, activism and courage as a source of strength to draw upon.
"I know that I will be sworn in tomorrow as the first, First Nations woman in the federal cabinet because of the shoulders there for me to stand on – shoulders like Lowitja's," she said.
Opening her speech in Wiradjuri, the language of her people, Ms Burney, revealed her own painful family history, where she did not meet her father until the age of 28.
Born 10 years before the successful 1967 referendum to allow the Commonwealth to make laws for Indigenous people and include them in the census, Ms Burney was raised by white members of her extended family because her mother was unmarried.
Ms Burney said landmark reforms for Indigenous Australians were introduced by Labor prime ministers, and she and Anthony Albanese wanted to add to that legacy by implementing the Uluru Statement from the Heart in its entirety.
"I was 10 years old when my country, when my fellow citizens, decided to count me in the population of the country I was born in ... Now imagine how this next referendum will make us feel," she said.
"About ourselves, our neighbours, and our country, when it passes."
Ending her speech, Ms Burney urged people to lend their "effort" to an enshrined Voice.
"My life has led me to this place and given me the chance to gather the country together on this task of nation building," she said.
"Lowitja did not fail in her task of building the nation and giving First Nations people a Voice.
"I promise you that I will not fail in this task and together, I know we will all succeed."
