The game looked to be heading for a stalemate, before Challinor struck twice in as many minutes seconds from time.
The result was harsh on Crawley, who had enough chances to wrap it up before the late double blow.
The best fell to striker Steve Burton just before half-time when he beat two defenders to create space for a shot but, having done the hard work, he missed the ball completely and ended up flat on his back.
The ball ran to Daryl Clare, whose curling effort was heading for the top corner before Santos Gaia headed off the line for a corner.
Defender Chris Giles rose highest to meet Simon Wormull's resulting flag-kick and this time Billy Jones was on hand to nod clear from underneath the crossbar.
Clare should have put the home side ahead on 74 minutes when he was sent through on goal by Burton and shot straight at keeper Paul Jones.
Craig Farrell was guilty of an equally poor finish for Exeter moments later when he outpaced the home defence to a long ball and gave Scott Ward a comfortable save.
The game swung in Exeter's favour following the introduction of striker Steve Flack.
They poured forward in the last ten minutes and got their reward 60 seconds into added time when Flack won a header in the box. It fell kindly on Challinor who squeezed in a shot from a tight angle.
Challinor made the game safe moments later when he pounced on a loose ball, rode two challenges and slotted under home keeper Ward.