Double defending champion Verstappen was leading Mercedes' Lewis Hamilton with a few laps left when Kevin Magnussen crashed his Haas into a wall, leaving debris on the track and halting the race.
The cars returned to the grid with fresh tyres for a two-lap sprint to decide the winner but in a chaotic restart, Williams driver Logan Sargeant drove into the back of AlphaTauri's Nyck de Vries to knock both of them into gravel at turn one.
Alpine drivers Pierre Gasly and Esteban Ocon then crashed into each other to leave a trail of debris on the track and trigger another red flag, the third and last of the race.
After lengthy deliberations, stewards decided the race would finish with one lap behind a safety car in the order of the previous start - minus the cars that had crashed out.
That allowed Verstappen to claim his second win of the season ahead of Hamilton with Aston Martin's Fernando Alonso third in a race in which eight of the 20 cars failed to finish.
Red Bull have now won all three grands prix in the new season, a record for the team, resuming their dominance after their runaway success in 2022.
"It was a bit of a mess but we survived everything .... and we won, which of course is the most important," Verstappen said after his first victory at Albert Park and Red Bull's first since 2011.
"That's a really good weekend for us, we have to be happy with that."
The Dutchman has 69 points in the championship race, 15 ahead of team mate Sergio Perez.
Alonso, third in all three races, is third in the standings on 45 points.
Red Bull, on 123 points, have opened up a big lead in the constructors championship ahead of second-placed Aston Martin (65) and third placed Mercedes (56).
ON FIRE
Mercedes' first podium of the season was soured with George Russell suffering his first retirement for the team after losing power and rolling to a halt with his engine on fire early on.
"It’s really unlucky for George today on our side," said seven-times world champion Hamilton.
"Otherwise, to get those points is really amazing today. I definitely didn’t expect to be second so I’m so grateful for it."
It was a dark day for Ferrari with Charles Leclerc knocked out of the race at the first turn after being clipped by Aston Martin's Lance Stroll.
His Ferrari team mate Carlos Sainz survived all the carnage and was in line to finish fourth but a five-second penalty for making contact with Alonso left him demoted to 12th and raging on the team radio.
Stroll took fourth instead in a big day for Aston Martin, with Perez placing fifth after starting his race from the pit lane following his disastrous qualifying on Saturday.
McLaren's Lando Norris was sixth ahead of Haas's seventh-placed Nico Hulkenberg.
McLaren rookie Oscar Piastri celebrated his first points with an eighth-place finish, while Alfa Romeo's Zhou Guanyu and AlphaTauri driver Yuki Tsunoda rounded out the top 10.
Verstappen had to recover from a poor start as he lost his lead to both Russell and Hamilton in the first three turns.
But after two early safety cars and a red flag when Williams driver Alex Albon crashed out, Verstappen reclaimed the lead by passing Hamilton at the 12th lap after Russell pitted for an early tyre change.
Verstappen opened up a big gap before the late red flags saw the race turn from procession to drama and back again.
Drivers and team bosses queried whether the red flags were necessary, or whether safety cars might have saved plenty of bother.
"Obviously, results like this are good for some teams and bad for others," Mercedes boss Toto Wolff told reporters.
"We just need to understand going forward when are red flags being pulled out and what is a safety car?"
