Rosberg set a best time of one minute 42.311 seconds around the Sochi circuit, which snakes past some of the landmark venues used at this year's Winter Olympics, in a sunny morning session with Hamilton second.
The Briton, who leads Rosberg by 10 points after winning the last three races, went top of the timesheets in an interrupted afternoon session with a best of 1:39.630.
"I like the new circuit; it's got really good grip and great corners, with a lot at medium speed. It's good fun," said Hamilton.
"The car has been feeling really good today. McLaren look close which is great for everyone as it should make for a decent race. We've got quite a good balance with the car so there are a lot of boxes ticked but we still have work to do."
Rosberg said he went down a different setup route to Hamilton in the afternoon and would revert to a more conservative one for Saturday.
The red flag came out eight minutes from the end when Australian Daniel Ricciardo's Red Bull had a power unit failure.
Mercedes are expected to wrap up the constructors' championship on Sunday with three races to spare, having won 12 of the 15 so far with eight one-two finishes. Hamilton, the 2008 champion, is chasing his ninth win of the season.
McLaren's Jenson Button was the early pacesetter as drivers got to grips with a layout compared to the former Valencia street circuit and Singapore, with the 2009 champion ending the morning as the third fastest.
He also collected two fines totalling 1,100 euros (866.80 pounds) for pitlane speeding and was sixth fastest in the second session.
Ferrari's Fernando Alonso was fourth and third in the two sessions, with Danish rookie Kevin Magnussen, Button's team mate, fifth and then second in the afternoon.
The only Russian regular, Toro Rosso's 20-year-old rookie Daniil Kvyat, was seventh in the morning and then eighth.
That was still ahead of both Red Bulls, his 2015 employers, with four times champion Sebastian Vettel only ninth.
Kvyat was joined for the first session by 19-year-old compatriot Sergey Sirotkin, making his debut at a race weekend in the Sauber usually driven by Mexican Esteban Gutierrez.
Williams had a disrupted opening session when a problem with a tyre blanket damaged a set of Valtteri Bottas's tyres.
All drivers took to the track with stickers of support for Jules Bianchi, Marussia's French driver who was severely injured in a crash in Japan last Sunday and remains critical.
"After what happened in Japan, you try and do the best you can. I would be lying if I said other things weren’t on my mind, but you do what you can to make sure you’re 100 percent committed to driving," said Ricciardo.
Marussia will have only one car running this weekend, with Bianchi's left in the garage as a mark of respect.
Chilton ended the first session with the slowest time but picked up speed in the afternoon.
"I don't know how to put into words how truly devastated I am by what has happened to Jules," he said. "It is going to be a very emotional weekend for the whole team, but we will try to get through it and keep praying for Jules."
(Editing by Peter Rutherford and Justin Palmer)
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