Pre-sale estimates for the rare machine expected it to sell for £40,000 to £60,000.
However, lively bidding at Bonhams in Knightsbridge saw it fetch £85,250.
Auction officials said there was a lot of interest with a large number of online and phone buyers as well as bidders in the saleroom.
A keen buyer from the US made the final bid and secured the historic machine.
Built by Heimsoeth and Rinke in 1941, the oak-encased machine which encrypted German codes during the Second World War is the three-rotor version, used between 1938 and 1944.
Patented by HA Koch at the end of the First World War, the device was intended for commercial purposes but by 1939 the majority of Enigma machines had been appropriated for German military use.
The construction of the world’s first top secret computing machine, Colossus, at Bletchley Park near Milton Keynes during the war meant the messages scrambled by Enigma could eventually be decoded in under 24 hours. The fact that the British managed to crack Enigma is credited with shortening the war as the Allies were able to know what the Germans were doing.