Galway are struggling this season and sit bottom of the Premier Division while the Candystripes are third after winning four of their five games.
The tie of the rounds sees Shamrock Rovers take on Bohemians in a Dublin derby.
League leaders Cork City visit Limerick, with the matches to be played on the weekend beginning 17 April.
Derry City will start their EA Sports Cup bid with a second-round game at Galway United later this month.