Edward was born in 1846 in County Louth, Ireland. After emigrating to the United States, he enlisted in the Army and was later deployed as a Private with Company F of the 4th US Cavalry to Texas. It was his actions on the 29th of September 1872, that would earn him the Medal of Honor. The citation reads:

Gallantry in action.

Edward received the Medal of Honor on the 19th of November of the same year and was discharged in January of 1876. It is unknown when Edward Branagan died or where his final resting place is, but there is a cenotaph for him at the Fort Concho National Historic Landmark in San Angelo, Texas.