David McGoldrick struck twice to help the home side continue their climb away from the foot of the table, taking his tally to 11 since joining on loan from Nottingham Forest.
John Fleck opened the scoring in the ninth minute before McGoldrick added his double either side of Billy Clarke's goal for high-flying Crawley.
"This result will make people think we have started to turn the corner," said Robins. "Crawley are a very decent side with some very bright players and they caused us problems with their movement.
"We are in a difficult run of games but our confidence is building and our fitness is improving. We lasted the 90 minutes very well. All our players contributed, I could go through the whole team. Some have contributed more than others but they have all put a shift in.
"They have done what we have asked of them, but we have to make sure that we keep doing it. I thought Crawley rode their luck with their goal. The ball smashed into the face of one of our players and fell to Billy Clarke. It was fortunate from their point of view."