Mercedes' George Russell was third on the slippery Albert Park circuit where only 14 of the 20 cars finished in the treacherous conditions.
Norris, the pre-season favourite for the drivers' title, started the Formula One season as he finished the last at Abu Dhabi, with victory from pole.
The Briton's win ended Verstappen's long run at the top of the championship standings dating back to May 2022.
Norris's teammate Oscar Piastri started second on the grid but finished ninth. He dreams of becoming the first home driver to claim victory or a consolation podium ending with a skid into the grass.
Norris claimed his first win in Melbourne and the fifth of his career, saying it was difficult with Verstappen breathing down his neck.
"It was amazing. Tough race, especially with Max behind me," he added.
--Reuters--