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The session was called off after just two of the three parts because conditions had deteriorated to the point that the track was too dangerous.
Ferrari's Sebastian Vettel took fifth but will be demoted 10 places.
Hamilton needs to beat Rosberg by two points and Vettel by nine to seal his third title this weekend.
The Red Bulls of Daniel Ricciardo and Daniil Kvyat will be on the second row with the Force India cars making up the third after Vettel's demotion.
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Vettel and team-mate Kimi Raikkonen, who qualified eighth, have been penalised for using more than the permitted number of engines.
Force India's Sergio Perez will start fifth ahead of team-mate Nico Hulkenberg.
The Ferrari drivers' penalties mean Williams's Felipe Massa, who qualified ninth, will start seventh, ahead of Toro Rosso's Max Verstappen and McLaren's Fernando Alonso.
"I felt comfortable with the car," Rosberg said. "It is quite challenging - the car snaps quite suddenly, but I got a good lap together."
Hamilton said: "It was lots of fun out there for all of us. It's a shame we couldn't go out at the end.
"I had hoped we could go out and clean the track. It looks like a lot of people have turned up today to watch. I hope the weather eases up for the race."
Ricciardo said he expected "we will have a lot of run" in the race.
The race is due to start at 14:00 local time (19:00 GMT) and weather forecasts suggest conditions will improve by then.
Austin, Texas, is still being affected by rain on the periphery of tropical storm Patricia and has seen record amount of rain in the last 24 hours.
Behind Alonso will be Williams's Valtteri Bottas, Lotus's Romain Grosjean and the second McLaren of Jenson Button.
Alonso has a new-specification Honda engine in his car, which was not available to Button this weekend, prompting the Englishman to joke on Thursday that he hoped his team-mate would "annihilate" him.
Unfortunately for him, he got his wish, at least in qualifying - Button was 0.928secs slower than Alonso, significantly more than the small improvement in power in the upgraded engine was worth in lap time.
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Button, who was complaining about a lack of rear grip, said conditions at the end of the second part of the session were too dangerous.
"At the end," he said, "the team were like, 'Come on, one last big push.' And I was like, 'Right'. I was just floating."
Vettel added that the weather had meant the track was in "red flag conditions".
A number of drivers had spins during the treacherous session, with the fast kink at Turn 10 proving the most dangerous part of the track - Hamilton, Vettel, Hulkenberg and Raikkonen were among the drivers who lost control there.
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But despite the difficult conditions only one driver crashed - Toro Rosso's Carlos Sainz lost control at Turn Four and smashed into the wall on his second flying lap in the first part of qualifying.
United States GP qualifying results
United States GP coverage details
Nico Rosberg beat Mercedes team-mate Lewis Hamilton to pole position at the United States Grand Prix as qualifying was truncated by rain.