Ajose scored the winner as the Reds came from behind to record their fifth victory in six games over Championship opposition.
Crawley's Kyle McFadzean headed against the crossbar early on but the Trotters took an undeserved lead when Benik Afobe, on loan from Arsenal, latched onto a pass from Martin Petrov and rounded Crawley's Paul Jones to score.
Nicky Adams and Jonathan Forte went close either side of half-time and Mat Sadler failed to connect properly with a header when well placed.
Bolton were denied a second when Marvin Sordell was flagged offside but Billy Clarke drew the home side level with a crisp, left-foot shot from 20 yards which gave Bolton keeper Andy Lonergan no chance.
Petrov and Forte went close at opposite ends before Ajose headed a Clarke cross past Lonergan in the 92nd minute to dump Owen Coyle's side out of the competition.