The flying Finn raced past Mercedes team-mate Hamilton on pole and led into the first corner before stretching away to win by more than 20 seconds around the Albert Park circuit.
Bottas also picked up the inaugural bonus point for setting the fastest lap as Britain's Hamilton battled to save second.
The 29-year-old Bottas was fifth in the championship last year after failing to win a race, but he drove with precision and determination Sunday to score his fourth grand prix victory and first since Abu Dhabi in 2017.
"I don't know what just happened. The start was really good, it was definitely my best race ever," Bottas was reported as saying on Reuters.
"I just felt so good and everything was under control. The car was so good today so truly enjoyable, I need to enjoy today."
Hamilton qualified on pole for the 84th time in his illustrious career but he was sluggish off the start and was unable to keep pace with this team mate.
Instead, the 34-year had to concentrate on fending off the Red Bull of Max Verstappen, who had moved into third after passing Ferrari's Sebastian Vettel.
"It was a good weekend for the team," said Hamilton. "Valtteri drove an incredible race today so he deserved it."
Ferrari struggled with its pace, with Vettel and Charles Leclerc trailing in fourth and fifth respectively.
Before the race, drivers and officials stood for a minute's silence to pay respects to F1 race director Charlie Whiting, who died Thursday aged 66, and for the victims of Friday's terror attacks in Christchurch, New Zealand.