The 30-year-old German started from pole and produced an imperious performance as he pulled clear of the field to take his fourth win of the 2016 season and the 18th of his career.
He won the final three races of 2015 and is now unbeaten since defending three time world champion Hamilton triumphed in last year's United States Grand Prix to secure the championship.
The only three men to have reeled off seven straight wins were Italian Alberto Ascari and Germans Michael Schumacher and Sebastian Vettel, whose race on Sunday was ended by a first lap accident.
Hamilton's race, however, was stymied by further mechanical problems.
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He had suffered an engine failure in qualifying on Saturday and he was forced to nurse his Mercedes home 25 seconds behind the rampant Rosberg.
Finn Kimi Raikkonen finished third in the second Ferrari ahead of compatriot Valtteri Bottas and his Williams team-mate Felipe Massa.
Fernando Alonso came home a much-improved sixth for McLaren ahead of Kevin Magnussen of Renault, Romain Grosjean for Haas, Sergio Perez of Force India and 2009 champion Jenson Button who was 10th in the second McLaren.
On a bright day on the shores of the Black Sea, with a track temperature of 41 degrees Celsius, the calm before the start was swiftly transformed into mayhem on the opening lap.
Rosberg made a near-perfect start, pulling clear from his 24th pole, but behind him cars collided in a dramatic incident that brought out the Safety Car before the race was a minute old.