The German, chasing his seventh successive win, will be joined on the front row by Finland's Valtteri Bottas in a Williams. Ferrari's Sebastian Vettel qualified second but has a five-place grid penalty.
Hamilton, the world champion and winner in Russia for the past two years but 36 points behind Rosberg after three races, did not take part in the final phase after a power unit problem. He is set to start 10th.
Mercedes said it was the same failure that the Briton had experienced in China two weeks ago, where he failed to set a lap time in qualifying and started at the back of the grid.
Hamilton had already been summoned to see stewards after qualifying for failing to follow the race director's instructions at turn two.