Harry Pritchard and substitute Danny Devine scored for the Bantams, with Reece Grego-Cox's late goal a consolation for the visitors.
Bradford began brightly and were rewarded in the 18th minute through Pritchard. Zeli Ismail burst away down the right and crossed for Pritchard to score with a first-time, left-foot shot from 10 yards.
Midfielder Devine scored the second home goal four minutes after coming as a half-time substitute for the injured Pritchard.
Devine exchanged passes with Aramide Oteh before cutting across the field and unleashing a left-foot shot from just outside the penalty area which flew into the net after taking a deflection.
Crawley always posed an attacking threat and Bantams keeper Richard O'Donnell did well to keep out a close-range left-foot shot from substitute David Sesay in the 70th minute.
However, the visitors had their reward in the 81st minute when Grego-Cox scored with a right-foot effort from just outside the penalty area and they caused Bradford some anxious moments as they pressed for an equaliser.
Match report supplied by PA Media.