The most obvious single battle that was influenced by ENIGMA may, ironically, be the (early stages of the) Battle of the Bulge. Hitler's paranoia had finally advanced to such state in 1944 that he was convinced the Allies were eavesdropping on his intelligence; so he insisted that all plans drawn up for that attack NOT use the cracked cypher. It is likely that the Allies had become used to never being grossly surprised, leading to a degree of over confidence that cost lives in December 1944.
Note that even though Hitler's conclusion was correct, this is still really just Hitler's paranoia showing though. The use of ENIGMA intelligence by the Allies was so sparing that he had no logical basis for this conclusion. Other than in the North Atlantic, where the battle was truly existential for the British, it was kept as a reserve to prevent exactly the scenario that occurred Dec. 16, 1944 - a massive German surprise attack in a weakly defended sector.
In the Allied favour the key battle/campaign would be that of the merchant marine convoys in the North Atlantic through the dark days of 1941 and 1942. The British never realized that the Germans had broken the merchant marine codes in the mid 1930's, so it was only with the help of ENIGMA intelligence that sufficient convoys got through to keep Britain afloat as it were.