Vettel was joined by fellow big-name drivers Kimi Raikkonen and Jenson Button in dropping out of qualifying in the second session as they struggled on the wet surface at the Albert Park street circuit.
While Vettel struck trouble, his new Red Bull teammate Daniel Ricciardo thrived, qualifying second, three tenths of a second behind Hamilton's pole time. Ricciardo narrowly missed becoming the first Australian to take pole in his home race.
Mercedes driver Nico Rosberg qualified third, ahead of McLaren's impressive debutant Kevin Magnussen, who put in a heady drive in tricky conditions, and Ferrari's Fernando Alonso.
Hamilton's pole position took him level on 32 career poles with British compatriot Nigel Mansell.
"Today was so much harder for everyone in these conditions," Hamilton said. "These cars are a lot harder to drive in the wet and the first time for me driving this car in the rain. To be up here is a great showing for the team."
Vettel had complained during the pre-qualifying practice session about the handling of the car, describing it as "horrible." The car's handling seemed to deteriorate even further during qualifying, with Vettel ending up 2.4 seconds off Ricciardo's pace in the Q2 session.
"I struggled with drivability," Vettel said. "We have lost that overnight, so we need to find out why that was."
Raikkonen had lagged behind his Ferrari teammate Alonso through the race weekend as they embark on what should be a keen rivalry, and his qualifying ended when he crashed into a track wall after hitting a wet patch when accelerating away from a bend.
McLaren's Button was caught out by the yellow flags from the Raikkonen incident and qualified 11th.
Button, Vettel and Raikkonen will all be promoted one spot on the grid because Williams' Valtteri Bottas was given a five-place penalty for changing a gearbox.
Ricciardo notched his career best in qualifying, bettering his fifth place at last year's British Grand Prix.
"I'm really excited but at the same time still staying calm and collected because tomorrow is what counts, so there's no point in having a party tonight," he said.
Rosberg's third position was less than he had hoped after topping the timesheets in pre-qualifying practice and in the Q1 session, but Mercedes had clearly been the fastest car over the weekend and he is still well positioned to fight for victory.
"It could have been better but third was OK and we have a good racing car, so from third anything can happen," Rosberg said.
Toro Rosso's Jean-Eric Vergne qualified sixth in another career-best performance — beating his seventh position at Canada last year.
Force India's Nico Hulkenberg was seventh, followed by Toro Rosso's 19-year-old debutant Daniil Kyvat in an impressive eighth and Williams' Felipe Massa in ninth. Bottas qualified 10th but drops down because of the penalty.
Sunday's race is forecast to be dry, with only a small chance of rain. It will be a journey into the unknown for all teams as they get a first chance to see how their new V6 turbo hybrid engines and reconfigured aerodynamics perform over a two-hour race.
Australia is among the highest fuel consumption races on the F1 calendar, and with fuel limits sliced to 100 kilograms this year compared to around 160 last year, Sunday's race looms as a test of engine and fuel management more than an outright race.