It was given to the Luftwaffe commander-in-chief for his 50th birthday by Erhard Milch in 1943.
Field Marshal Milch, who was half-Jewish and received Goering's protection, was found guilty of war crimes after World War Two.
The box was sold to an anonymous phone bidder by J P Humbert Auctioneers.
It was one of a number of items associated with Goering sold at the auction in Whittlebury in Northamptonshire.
A silver bowl presented to him by the German Hunting Association went for £2,220 and an ivory letter opener given to him by the Japanese government made £680.
Auctioneer Jonathan Humbert said the results "show the continued fascination with high-ranking members of the Nazi Party".
A cigar box once owned by Nazi number two Hermann Goering has sold for £3,200, nearly the auctioneers' estimate of £800 to £1,200.