Rovers made a perfect start as Paul Wanless netted his sixth goal of the season just two minutes into the game.
The former Oxford United midfielder cracked home a long-range drive to continue his superb start to life at The Lawn.
Just three minutes later Alex Meechan made it 2-0 as he converted Scott Rendell's ball in from the left, leaving Town keeper Phil Smith with no chance.
Rendell almost headed a third within the opening ten minutes, before the visitors managed to stem the tide without troubling Ryan Clarke in the Rovers goal.
Rendell and Abbey both tested Smith just past the half-hour mark, with Jamie Cade's shot a rare effort from the Reds.
Abbey headed a Meechan cross against the woodwork to end a first half of total Rovers domination.
Crawley brought on Steve Burton at half-time, while the injured Wanless had to be replaced by Scott Rogers.
Burton soon headed narrowly over, before Daryl Clare tested Clarke as Crawley looked for a way back into the game.
Twenty-five minutes from time it was almost game over as Meechan crashed a 25-yard effort against the frame of the goal with Smith well beaten.
Sub Simon Wormull drove a 20-yarder just over the crossbar and David Woozley headed wide from a free-kick, before Clarke pushed away another effort from Clare.
Clare then set up a tense finale by converting from the spot six minutes from time after Adam Garner had brought down Burton.
Unbelievably, Town then snatched a point as Sacha Opinel's cross picked out Ben Judge, who netted from close range to complete the comeback.