Billy Clarke took Crawley into the top six by scoring twice in a second half of five goals and two dismissals.
Danny Kedwell's penalty put the Gills ahead after Kyle McFadzean fouled him.
Crawley's Gary Alexander and Gills' Charlie Lee then each saw red for second yellows for respective fouls on Leon Legge and Nicky Adams.
McFadzean equalised from close range and Clarke was on target with a header and an individual effort either side of a close-range Legge leveller.
VIEW FROM THE DRESSING ROOM
Crawley manager Richie Barker told BBC Sussex:
"We've made more hard work of it than should have been and that's no disrespect to Gillingham. I thought we were the best side and we should had won the game more comfortably.
"To come from behind, certainly at 1-0 down and down to 10 men, we could have easily gone under.
" I think we would probably have taken this position at the start of the season bearing in mind what start we had. We won't be getting too carried away."
Gillingham boss Martin Allen told BBC Radio Kent:
"I thought it was a good performance, a couple of set-plays have cost us dear. I'm disappointed to lose the game.
"The sending-off of Charlie Lee was crucial. Charlie Lee was shielding the ball off of play, it wasn't even a foul, an unbelievable decision.
"We have to be careful not to use the word 'desperate' but obviously they [the players] want to turn these good performances into victories and to concede from two set-plays is disappointing."
Crawley Town 01 P Jones 02 Hurst 03 Sadler 04 Connolly 05 McFadzean 10 Drury (Proctor - 72' ) 11 Simpson 15 Bulman 16 Adams (Torres - 87' ) 07 Clarke Booked (Sinclair - 90' ) 09 Alexander Dismissed after an earlier booking Substitutes 25 Maddison 06 Essam 08 Torres 18 Rooney 20 Sinclair 14 Proctor Gillingham 01 Nelson 03 Martin 06 Legge (McKain - 77' ) 18 Lee Dismissed after an earlier booking 26 Barrett 07 Whelpdale Booked 12 Gregory 15 Dack 19 Hollands (Linganzi - 63' ) 08 Akinfenwa (Weston - 32' ) 09 Kedwell Booked Substitutes 21 Howard 22 Butcher 04 Linganzi 11 Weston 14 Allen 16 German 25 McKain