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"He has shown great promise so far in his debut season and I know he will build on this with every race," said the team owner of Rosberg.
"I must also commend Mark Webber for his contribution to the team over the past two seasons as he has been a credit to himself and Williams and has never let up in his pursuit of competitive success."
Williams had an option to continue with Webber, who joined from Jaguar in 2003, for two more seasons but let it lapse on Monday.
The Australian had hinted strongly that the stumbling point was money, with Williams having to pay him significantly more under the terms of the deal.