The 55-year-old has signed a two-year contract with the Red Devils, who have an option to extend it by a further year if they wish.
BREAKING NEWS: We are delighted to announce the appointment of Dermot Drummy as our new Head Coach. pic.twitter.com/IAlcF4gPOI— Crawley Town (@crawleytown) April 27, 2016
"Hopefully I can produce a team that plays attractive football that excites our fans and brings the whole club together – players, supporters and the people who work for the club," Drummy told the club website.
"I can't wait to get started. I'm looking forward to meeting the players and preparing for Saturday's game."
Drummy has appointed former Aldershot coach Matt Gray as his assistant.