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 1 season as he finished the last at Abu Dhabi, with victory from pole.
His teammate Piastri started second on the grid but finished ninth, his dreams of becoming the first home driver to claim victory or a consolation podium ending with a skid onto the grass.
"It was amazing. Tough race, especially with Max behind me," said Norris.
"I was pushing, especially in the last two laps. It was a little bit stressful, not going to lie.
"This time we got it right and ended on top so I’m happy."
Red Bull kept Verstappen out on worn tyres hoping for a break in the rain but he was ultimately forced to pit, potentially costing him a better shot at taking out Norris. He said it was "worth the gamble".
"It was quite spicy on the slick tyres. It was alright in the end," he said.
"This is where I expected to be. In the first stint we were lacking a bit of pace to the McLaren."
Lewis Hamilton's debut for Ferrari proved a damp squib with the Briton finishing 10th, two places behind teammate Charles Leclerc, but it was an encouraging day for Williams with Alex Albon finishing fourth after Mercedes' 18-year-old rookie Andrea Kimi Antonelli was demoted to fifth due to a five-second penalty for an unsafe release from the pit.
The race was held up for 15 minutes after debutant Isack Hadjar lost control in the formation lap at turn two and rear-ended his Racing Bulls car into a wall.
With his rear wing bent out of shape and his race over before it began, the unharmed Frenchman was in tears as his car was hauled away by a crane.
While Norris had a clean start, Piastri was skidding left and right at the first turn, allowing Verstappen to roar past him on the outside.
Australian rookie Jack Doohan was gone soon after the restart, crashing his Alpine at turn six on the first lap to trigger the safety car as his father Mick, the motorcycling great, looked on in disbelief from the team garage.
The drama continued as Williams driver Carlos Sainz spun out of control at turn 14 during the safety car's deployment.
When the track was cleared of danger, Norris held Verstappen at bay as Piastri nipped at the Dutchman's heels.
The front three's order held until lap 16 when Verstappen locked a front brake and veered wide to allow Piastri to snatch back second.
Piastri, with more pace than his teammate, had Norris in his sights but was ordered to hold position by the McLaren garage until a rain shower passed and the track dried.
Twice world champion Fernando Alonso became the fourth car knocked out when he veered off track at turn seven on lap 34 and crashed, bringing out the safety car again.
Norris and Piastri pitted to change to hard tyres but another rain shower wreaked havoc 10 laps later, as Piastri skidded into gravel then ended up in the grass. He managed to reverse out and re-enter the track but his race was effectively over.
Red Bull's Liam Lawson and Sauber's Gabriel Bortoleto then crashed out in quick succession, triggering a third safety car deployment.
Though that gave Verstappen a last crack at reeling in Norris, the Briton showed great composure to hold the relentless Dutchman at bay and land the first blow in the championship ahead of round two in China.