"Eugene was a happy, kind and caring boy and he loved nothing more than to dance. Eugene was loved by so many people and he brought us, his brother and four sisters, extended family, friends and community so much joy and happiness in his short life."
He loved nothing more than playing with his siblings and dogs, they said.
"We could not have been more proud of him and cannot imagine our world without him."
The family have asked for privacy.
Tributes are pouring in online for Eugene.
Steph Randall said the boy was a "beautiful little man" while Alikki Kostakakis was "completely numb and lost for words".
"[Eugene] was and always will be an amazing little man. God took him too soon," she said.
WorkSafe Victoria said last week it was investigating the incident, including whether height requirements for all riders had been followed.
Anyone on the spinning amusement ride should be over 130cm, according to the Wittingslow Amusements website.
Initial inquiries indicate the ride may have been used by carnival workers and their families after the carnival had closed, WorkSafe said.
In a statement, the carnival operator said: "Our hearts and sincerest thoughts remain firmly with his family at this sad time, as well as their friends and extended family here and in New Zealand.
"We grieve with them and we are doing everything we can to ensure everyone gets the support they need. The entire Wittingslow family and all of our staff are devastated by what occurred last week."
