Referee Stuart Attwell took the decision after initially taking the players off at the Checkatrade.com Stadium to assess the worsening conditions.
A third-round meeting with Birmingham at St Andrew's was at stake for the winner and, in wet and windy weather, Crawley striker Jamie Proctor was thwarted early on by goalkeeper Steve Mildenhall in front of goal following a cross by Nicky Adams.
A free-kick from Rovers defender Lee Brown sailed narrowly wide midway through the first half when Town 'keeper Paul Jones and Kyle McFadzean left the ball for each other.
Proctor headed a Billy Clarke cross wide before Joe Walsh came close to breaking to deadlock when his goalbound header was pushed on to a post by Mildenhall.
Crawley striker Emile Sinclair headed a Mark Connolly cross over shortly after the break, with Jones having earlier tipped a teasing cross from Michael Smith over.
But, with extra-time potentially on the horizon, referee Attwell decided to abandon the replay and the two sides will have to go again for a third time.
League Two Bristol Rovers held Crawley to a goalless draw at the Memorial Stadium in the initial meeting between the two sides earlier this month.