The home side began the game well and could have gone ahead through Kyle McFadzean, who rose highest in the area from a corner, but his header went just past the post.
Crawley did take the lead when Jamie Procter found the net in the 26th minute as he deflected Andy Drury's shot past Simon Moore.
City threatened just before the break when Wade Elliott drove an effort just over the bar from 20 yards out.
After the break, Proctor came close to adding a second for his side when he drilled a shot at goal from inside the box, but Moore was on hand to make the stop.
Matt Tubbs then came close for the home side as he managed to find space on the edge of the area, but he curled his effort just past the post.
Both sides pressed for a winner late on, but were forced to settle for the point.