Nick Tsaroulla broke the deadlock with a remarkable solo-goal, early in the second-half.
Ashley Nadesan doubled the Crawley lead, as his low drive went through the legs of Kiko Casilla.
Crawley’s incredible afternoon was complete, when Jordan Tunnicliffe turned the ball into the roof of the net following a free-kick.
Jack Powell curled an effort wide from the edge of the box, as Crawley made a strong start to the game, before Ian Poveda teased in an inviting cross which Rodrigo couldn’t connect with from close range.
Pablo Hernandez showed good feet before firing wide from 20 yards, then a good save from a narrow angle saw Glenn Morris keep out Poveda.
Kalvin Phillips mistimed a pass, which was pounced on by Nadesan but after darting into the box his effort was blocked.
From the resulting corner Crawley came so close to scoring, as Tom Nichols met the cross strongly with his head, but a brilliant close-range save from Kiko Casilla kept the score level, palming away from the top corner.
In the second-half it was the Reds who broke the deadlock through a piece of individual class from Nick Tsaroulla, who wriggled through two Leeds players, before breezing into the box past numerous challenges and drilling into the bottom corner. A stunning first senior goal, scored in the 51st minute.
Things got even better for Crawley moments later, when Nichols slid the ball through to Nadesan, who fired the ball through the legs of Casilla, to enhance Crawley’s chances of an upset on the 53rd minute.
Crawley’s FA Cup fairy-tale was becoming a reality when, from a free-kick, Nadesan’s close range volley was kept out by Casilla, but fell to Jordan Tunnicliffe, who poked the ball into the roof of the net from close range, to put the Reds three goals up, in the 71st minute.
Max Watters had a great chance to make it four, when he chased a loose pass and beat Casilla to the ball, but he couldn’t get his shot off in time as it was cleared off the line, the last action of an unbelievable afternoon for Crawley Town FC.
Crawley: Morris, Francomb, Craig, Tunnicliffe, Dallison (M. Wright 90’), Tsaroulla (J. Wright 71’), Powell, Hessenthaler (Doherty 90’), Matthews (Davies 71’), Nadesan (Watters 71’), Nichols
Leeds: Casilla, Davis (Greenwood 55’), Struijk, Cooper (Jenkins 45’), Alioski, Phillips, Shackleton (Casey 45’), Hernandez, Poveda (Raphinha 55’), Costa, Rodrigo (Harrison 45’)
Report by Coren Blackburn