They say he was inspired by the Islamic State (IS) group.
Officers say the attacker repeatedly stabbed a 59-year-old man in a suburban park and then tried to stab a policeman.
A large knife has been seized from the accused, who will appear in court later on Sunday.
The victim, who was reportedly walking his dog in the park, suffered injuries to his body and hands, and remains in hospital in a serious condition.
'New face of terrorism'
Although the authorities say they believe the attacker was inspired by IS, they do not think he has close ties to any extremist organisation.
"This is the new face of terrorism. This is the new face of what we deal with," New South Wales state police Deputy Commissioner Catherine Burn said.