The 61-year-old died last Friday, September 21 three days before he was due to face court charged with 21 child sex offences dating back 30 years.
Details of the allegations became public this week, prompting Prime Minister John Howard to ask the family to reconsider his offer of a state funeral, Australian Associated Press reported.
Addressing a packed St Mary's Cathedral, Fr Brennan said it was a time for remembering a gifted politician, a loyal friend and a dedicated family man, who was a father-of-three.
"He never sought honour or wealth,'' Fr Brennan said.
"He loved the mix of the personal and the political, and he idolised his heroes in the Labor party.
To Mr Collins' family Father Brennan said: "He never wanted to leave you with any mess.
"God alone is our judge and God alone is Bob's judge. This is not a day for judging Bob by his political opponents or his accusers."
Despite the controversy surrounding the allegations of child sex abuse this week, several members from the Northern Territory Government attended the service, including Chief Minister Clare Martin.