Leclerc became the first driver other than Lewis Hamilton to start first on the Melbourne grid since 2014.
Sergio Perez in the other Red Bull took his spot in the second row after coming in third ahead of McLaren's Lando Norris.
There was better news for struggling Mercedes as they took fifth and sixth through Lewis Hamilton and George Russell.
After claiming his second pole from the opening three rounds, Leclerc said: "It is a track where I have always struggled in the past but we worked hard and we got everything together. I am very happy to be on pole.
"The car is nice to drive. We just need to do a good start and hopefully we can keep that position."
Verstappen, the winner in Jeddah a fortnight ago, said: "I have not felt good in the car all weekend.
"There has not been a lap where I have been confident so it has been a bit of a struggle."
Hamilton's bid for a record eighth crown has been derailed by his uncompetitive Mercedes machinery. The British driver is already 29 points behind championship leader Leclerc.
But after he qualified only 16th at the last round in Saudi Arabia before taking a sole point for finishing 10th, Hamilton will take some comfort from his improved grid slot in Australia.
Qualifying was briefly suspended following a clumsy collision between Nicholas Latifi and Lance Stroll.
Stroll was caught unawares as the Williams driver made a move on his inside at Turn 5 and Latifi was sent spiralling into the wall.
Earlier, Sebastian Vettel was fined €5,000 after riding a moped on the track at Albert Park in violation of the rules, as his season went from bad to worse.
The four-time world champion jumped on a scooter to make his way back to the paddock after abandoning his broken down Aston Martin, which had smoke belching from the rear, in the first practice session on Friday.
The German, 34, was seen casually driving around the track, smiling and waving to fans, after the other cars had returned to the pits.
At one point he took his hands off the handlebars.
"This is the most iconic end to a practice session we can remember," Formula One's official Twitter account posted.
However, governing body the FIA took a dim view and found him in breach of sporting regulations "which prohibit anyone from being on the track in the five-minute period after the end of a session, with the exception of specifically identified personnel."
In a major renovation to the Melbourne track, five corners were re-profiled at Albert Park and two were removed completely.
After two races, Leclerc leads the drivers’ championship with 45 points, followed by Sainz on 33 and Verstappen on 25.