Defending champion Max Verstappen's Red Bull will start third on the grid ahead of fourth-placed Mercedes driver George Russell but it was a disappointing session for Ferrari with Charles Leclerc to start seventh and Lewis Hamilton eighth.
Briton Norris lapped the lakeside circuit in 01:15.096 seconds, 0.084 ahead of Piastri as the final session of qualifying went down to the wire.
"It's the perfect way to start the year," said Norris.
"I’m never going to get ahead of myself, I’m not that kind of guy. I’m confident the car is in a great position but we’ve also never run in the wet."
Rain forecast for Sunday could make the race a lottery but McLaren were delighted with their raw pace in the dry.
Verstappen, bidding for a fifth consecutive drivers' title, had struggled with handling problems through free practice so he was also glad to be near the front for Sunday.
"Yesterday was quite tough so for us to be in P3 today I will take that," he said.
"For me dry or wet is fine, in the weather there is always some crazy things that can happen and especially around here it can be quite slippery."
Yuki Tsunoda was fifth for Racing Bulls and drove like a man with a point to prove after missing out on a race seat at Red Bull to New Zealander Liam Lawson.
Williams also landed a blow in the midfield battle, with Alex Albon sixth, four places better than new teammate Carlos Sainz.
The Williams man was just ahead of the Ferraris, whose underwhelming pace was a surprise to their rivals and to Hamilton; his quickest lap in the final qualifying session was nearly nine-tenths of a second slower than Norris's. He also lost control and spun in the middle of the track at one point during Q2.
"I didn't know we'd be nine-tenths off or eight-tenths off today but there is a lot to dissect, for sure," Hamilton said.
"Tomorrow's going to be a challenge, I've never driven this car in the rain. I don't even know the rain settings so I've got to go and study that tonight."
Pierre Gasly was ninth for Alpine, comfortably out-qualifying his new teammate Jack Doohan (14).
Racing Bulls rookie Isack Hadjar was 11th, the best of the six drivers in Australia starting a Formula 1 season for the first time.
It was a mixed bag for the others, with Sauber's Gabriel Bortoleto 15th after scraping into the second session of qualifying (Q2).
Having triggered two red flags in two days of practice, Haas rookie Oliver Bearman suffered a gearbox problem in his out-lap and had to return to the garage, unable to post a time.
It was a similarly fruitless day for Red Bull's Lawson, who was knocked out of Q1 after sitting out the final practice earlier due to a power unit problem.
Mercedes' Andrea Kimi Antonelli also missed Q2, knocked out of the top 15 by a last-ditch lap from Bortoleto.