John Joe-O'Toole hit a dramatic stoppage-time winner as Rovers came from behind to clinch a clash with the Championship side.
O'Toole edged his side through in the first minute of added time with a low shot at the near post.
Crawley started well and went ahead on 14 minutes when Nicky Adams crossed and Jamie Proctor headed in at the far post.
Rovers threatened when a cross by Michael Smith was headed on by David Clarkson and keeper Paul Jones made a vital save.
Jones did well to tip over an effort from Elliot Richards following a corner by Lee Brown.
But Richards made it 1-1 in the 83rd minute with a right-foot shot and Tom Parkes then hit the post for Rovers with an overhead kick.
And the League Two side, who played with increased confidence in the second half, continued to create chances and O'Toole struck late on to clinch a