Connolly sent a towering header past Bartosz Bialkowski in the 53rd minute from Nicky Adams' corner and if it had not been for their goalkeeper struggling County, who have only won once on their travels this season, would have suffered an even bigger defeat.
Former Southampton keeper Bialkowski made a string of superb saves and when he was beaten the woodwork came to the visitors' rescue as Jamie Proctor hit the crossbar with a header before striking the post with a low drive.
Notts offered little, although defender Alan Sheehan hit the post in the seventh minute from a corner.
But Sheehan's afternoon ended on a sour note when he was sent off with 12 minutes to go for deliberate handball, having been booked earlier in the second half for a foul.
Crawley dominated for long spells as they claimed their second win in three with Mike Jones, skipper Andy Drury and Emile Sinclair all going close as John Gregory enjoyed his first home win since taking charge at the beginning of December.