Sotheby's auction house said that the score, which was discovered in a private collection in 2004 having been presumed lost after its 1908 St Petersburg premiere, was bought by a private collector.
It is thought to be the sole surviving handwritten manuscript of the symphony, and has been on display at the British Library in London since 2005.
The symphony, in E minor, was created in Dresden for a set of concerts organised by Rachmaninov's cousin.