Cheltenham maintained their unbeaten start with a 0-0 draw but needed goalkeeper Trevor Carson to be at his best to deny Crawley.
The Irishman pulled off an excellent save in each half to keep the game all-square and continue his good form since signing from Bury in the summer.
Carlisle dominated the first half chances and the pick of those came in the 25th minute when Patrick Brough dragged the ball back from the byline for debutant Jack Marriott whose low shot forced Carson into a superb reflex stop.
Cheltenham's best effort of the half fell to Lee Vaughan four minutes later but, after cutting in off the right flack, his low drive lacked the power to trouble visiting keeper Mark Gillespie.
Brad Potts and Danny Grainger squandered further chances for Carlisle before the interval and the Cumbrians continued to edge the proceedings after the break with Gary Dicker heading wide.
Marriott, on loan from Ipswich, should have snatched the winner in the 68th minute but after racing clear of the Robins defence he was foiled by another excellent Carson save.
Substitute Koby Arthur had a late chance for the home side but dragged the ball wide.