The German, who is chasing his seventh win in a row and fourth of the season, lapped Sochi's Olympic Park circuit with a best time of one minute 38.127 seconds set on the supersoft tyres.
Triple world champion Hamilton, the only driver to have won in Russia since the race's debut in 2014 but now 36 points behind Rosberg with 18 rounds remaining, was second fastest in 1:38.849.
The Mercedes pair, who have won 35 of the last 41 races between them, were comfortably ahead of the rest with Rosberg more than a second faster than Sebastian Vettel's third placed Ferrari, although the latter was using slower soft tyres.
Ferrari's Kimi Raikkonen was fourth in 1:39.332 with the Mercedes-powered Williams of Brazilian Felipe Massa fifth despite spending much of the session watching his mechanics working on the car.
Australian Daniel Ricciardo, who tested a new head protection device at the start of the session, was sixth for Red Bull.
Russian test driver Sergey Sirotkin, who will be competing again in the GP2 support series this season, replaced Denmark's Kevin Magnussen for the session and lapped nearly a second faster than regular race driver Jolyon Palmer in the other Renault.
Sirotkin ended up 13th fastest with Palmer 18th.