The Reds led 3-1 with 26 minutes to go following goals by Gary Alexender in the first half and two in quick succession after the break scored by Nicky Adams.
However Cheltenham scored two goals in the space of 11 minutes through Terry Gornell and Byron Harrison.
The eualiser forced both sides to play 30 minutes of extra-time during which the hosts dominated.
Finally with six minutes to go Harrison scored a fourth goal to seal defeat for Crawley.
Alexander gave them the lead with a stooping header in the 23rd minute after an excellent cross from the right hand side byling by summer signing James Hurst.
Matt Richards put Cheltenham back on level terms from the penalty spot five minutes before half-time.
The spot-kick was awarded when Mat Sadler brought down striker Byron Harrison who was put through from the right by Sam Deering.
Reds boss Richie Barker named an unchanged side from that which beat Coventry City 3-2 in their opening league game on Saturday.
Luke Rooney was named on the bench after signing this afternoon on loan from League 1 rivals Swindon Town.
Billy Clarke was also on the bench having missed the opening game through the birth of his baby boy.
Cheltenham made one change to the side which drew 2-2 with Burton Albion. Harrison came in for the injured striker Jamie Cureton.
Cheltenham came out the fastest at the start of the second half as Russell Penn seized upon a poor back pass by Kyle McFadzean and blasted a shot against the crossbar before Gornell followed-up with a shot over the bar.
Deering then had another chance for the hosts which flew wide from outside the area.
Crawley then raced to a dramatic 3-1 lead with two goals in six minutes by Adams.
The first of these in the 58th minute rebounded just inside the left post after a deflected challenge by Deering.
The second was more skilfully put away as Adams was found with a great cross by Jamie Proctor and he rounded keeper Scott Brown with a dummy before placing into the net.
However the game was far from over as the Robins recovered to 3-3 within 11 minutes with two goals by Gornell after 69 minutes and Harrison in the 75th.
Gornell netted at the second attempt after Paul Jones saved his first effort.
Reds’ skipper Josh Simpson then had a low shot well-saved by Brown from 20 yards.
Harrison put the hosts back on level terms with a well-judged header following a corner.
It was then end to end stuff as first Sadler tried to find Mike Jones in the area but the chance was cleared, then substitute Billy Clarke, who had come on for Proctor, was denied by Brown with a curling shot from 18 yards.
REDS: Paul Jones, James Hurst (Luke Rooney 101), Mat Sadler, Kyle McFadzean, Gary Alexander, Josh Simpson, Joe Walsh, Jamie Proctor (Billy Clakre 70), Dannie Bulman (Andy Drury 78), Nicky Adams, Mike Jones.
Unused subs: Mark Connolly, Connor Essam, Sergio Torres, Johnny Maddison
ROBINS: Scott Brown, Troy Brown, Steve Elliott (Keith Lowe 103), Sam Deering (Ashley Vincent 101), Byron Harrison, Terry Gornell, Jermaine McGlashan (Zack Kotwica 108), Matt Richards, Russell Penn, Sido Jombarti, Craig Braham-Barrett
Unused subs: Billy Jones, Jason Taylor, Connor Roberts, Joe Hanks
Referee: Oliver Langford
Attendance: 1,562 (117 away)