Ben Clark will take charge of first team matters on an interim basis for the immediate future, the Heed have announced. He will be supported by experienced Jacob Butterfield, injured midfielder Greg Olley and Luke Clark.
The players produced a perfect response following Elliot's exit on Tuesday, winning their National League Cup tie with Middlesbrough last night.
The 38-year-old officially replaced Mike Williamson in the summer after he left to join Milton Keynes.
Elliot took over the team on a caretaker basis almost a year ago, achieving a sixth place finish and claiming FA Trophy glory at Wembley against Solihull Moors in May.
The Heed sit fifth in the National League table but the former Newcastle goalkeeper moves to the Sussex club, who have taken seven points from seven League One matches.
Moving to Crawley with him are assistant Louis Storey and coach Anthony Sweeney.
Chairman Neil Pinkerton said: “Rob, Louis and Antony depart with the best wishes of everyone associated with Gateshead FC.
"Rob and Louis have been an integral part of the club’s success in recent years, culminating in an unforgettable day at Wembley Stadium in May which saw them lift the FA Trophy for the first time in our history.
"Thought we are saddened to see Rob, Louis and Antony leave the club, we understand that the approach represented an opportunity for the trio to progress their coaching careers at a higher level.”
Gateshead’s next match is Saturday’s game with struggling Wealdstone at the International Stadium.