Britain's Scott Redding was third on a Ducati in the 250th race of the MotoGP era.
Spaniard Marquez stretched his championship lead over compatriot Jorge Lorenzo, who finished 10th, to 24 points. Marquez has 145 points to Lorenzo's 121 and Rossi's 103.
Miller was the first Australian MotoGP winner since Casey Stoner at Phillip Island in 2012 and the first on a non-factory bike since Spaniard Toni Elias in 2006.
"I don't know what to feel at the moment," said an emotional Miller, his voice croaking, after crossing the line with the bike's front wheel high in the air.
"A lot of people have badmouthed us and everything like that, and said that this project wouldn't work. And I hope we've shown them wrong and that we can ride a bike, that I'm not an idiot," said the non-works Marc VDS Honda rider.
"Thank you for Honda for taking this risk on me."
The 21-year-old Australian, whose nickname 'Jackass' is written on the back of his helmet, came into MotoGP last year with a reputation for aggressive riding and bold antics.
Italian Andrea Dovizioso started on pole for Ducati but the race was halted after 14 laps due to torrential rain and standing water.
It re-started after conditions eased for the final 12 laps, with Dovizioso crashing out from second place as Rossi took charge.
The Italian great's hopes of repeating last year's win at a favourite track did not last long, with Rossi skidding into the gravel a lap later.
Marquez then led, with a string of other riders falling, until Miller made his move.
"Today was the race to lose the points," said Marquez. "I was really quiet, really concentrated.
"I saw that Valentino was pushing a lot, I said 'Okay, I cannot follow him'. When I saw he crashed, I saw that Miller was behind me.
"When he passed me, I just said 'this second place today is like a victory'. It was really critical conditions and these 20 points will be really important in the end for the championship."
(Reporting by Alan Baldwin, editing by Clare Fallon)
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