Well.... yes and no. 

Actually, most movies or stories about the Enigma cracking don't give credit where credit is due - there was a group of about eight Polish mathematicians who had done a LOT of groundbreaking work on cracking Enigma before 1939. These Poles escaped to the UK when the Nazis invaded Poland, and they brought their work with them. This gave the Brits a leg up on the project.

The "computer" used in cracking Enigma wasn't really a general purpose computer - it was more of a combinatoric eliminator that was both mechanical and electrical. The first electronic computer used in codebreaking was the COLOSSUS, it was built with enough vacuum tubes to heat a modern high-rise apartment building, and that was used to break the Lorenz code for the teletype circuits of the German high command for strategic communication. That was a pretty tough code to break as well.