All the goals and a sending off apiece for both sides came in an eventful second half.
The hosts carried more of a threat in the first half and Gillingham goalkeeper Stuart Nelson did well to smother a goal-bound shot from experienced striker Gary Alexander following a deep cross by James Hurst.
The Gills were relieved when an angled shot from Clarke was palmed away by Nelson after the former Blackpool midfielder got into a dangerous position.
The stopper later went down smartly at the near post to save a low shot by Nicky Adams.
Gills boss Martin Allen made a change after only 32 minutes by withdrawing uninjured striker Adebayo Akinfenwa in favour of Myles Weston, and the substitute went close shortly before the break by firing a low drive wide.
Danny Kedwell gave the visitors the lead from the penalty spot two minutes into the second half after he was fouled by Kyle McFadzean.
Alexander, booked earlier for a foul, was sent off in the 51st minute for a kicking Leon Legge and Gillingham's Charlie Lee received his marching orders two minutes later for upending Adams - his second bookable offence.
McFadzean rammed the ball home from close range to draw Crawley level moments later after Clarke headed on a Mat Sadler free-kick, and Clarke fired them ahead with a close-range header when an Adams free-kick came back off the post.
Legge hit back for the Gills to poke the ball in following a cross by Bradley Dack, but Clarke clinched victory for the hosts nine minutes from time with a fine individual strike.