Rosberg took his third successive pole position, with Hamilton alongside on the front row, in a qualifying session postponed from Saturday and then cut short by further bad weather.
Hamilton will become a triple champion with three races to spare if he scores two points more than Rosberg and nine more than Ferrari's Sebastian Vettel, who qualified fifth but has a 10-place grid penalty.
The grid places were decided by the positions in the second of the three qualifying sessions after the final phase was delayed and then cancelled.
"Qualifying was cut short so I guess we didn't really get to battle it out," said Hamilton, who was fastest in Saturday's final practice and would have been on pole had qualifying been cancelled altogether.
"But qualifying is just the beginning. The race is obviously a long battle and it's going to be hard out there today with those conditions varying throughout the race.
"There's no need to take any particular risks...it's not do or die this weekend," added the 30-year-old, who looks set to become the first Briton to win back-to-back titles.
Rosberg, who failed to convert his last two poles into victories and has won just three races to Hamilton's nine so far this season, was delighted to have repeated his 2014 pole and given himself the best shot.
Hamilton won that race in Austin last year, however, and is chasing a hat-trick of wins at the Texan circuit having also triumphed in 2012.
Both of his previous two victories in Austin came from second on the grid.
"It's been a really good qualifying and I felt comfortable in the car," said Rosberg after Mercedes's 12th front row lockout in 16 races.
"It's difficult out there with aquaplaning and everything because when it happens you just really struggle to have control over the car."
Australian Daniel Ricciardo will line up in third place for Red Bull with Russian team mate Daniil Kvyat alongside.
Mexican Sergio Perez will move up to fifth for Force India and has German team mate Nico Hulkenberg next to him.
The qualifying was red flagged in the first session when Spaniard Carlos Sainz hit the wall in his Toro Rosso and resumed after a 10 minute break.
"It was because of standing water and a used tyre. I'm not sure I will make it into the race," said Sainz, who crashed heavily in final practice in Russia two weeks ago but made it back for the race.
Vettel just made it through to the second phase after brushing the wall and ending up 15th on the timesheets.
A succession of drivers, including Rosberg and Hamilton, then spun at the Turn 10 with four times champion Vettel saying over the radio that he felt the conditions were so bad the session should be stopped.
The fans at least got to see some action for the first time since Friday, with Saturday's final practice run behind closed doors for safety reasons.
"It was good for the fans to get to see the cars running," said Hamilton. "It seems like there's a big crowd there today so I hope the weather backs up a little bit for the race."
(Reporting by Alan Baldwin, editing by Toby Davis and Pritha Sarkar)
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