The Hornets fought valiantly for 69 minutes before Brad Young fired the Blues in front. Zach Clough confirmed Carlisle's passage into the second round late on.
But who stood out for Horsham? Here are Sam Pole's ratings from the game.
1. Sam Howes - 9
Imperious between the sticks. The Horsham stopper made a number of key saves to allow the Hornets to stay in the game for as long as they did. Photo: Adam Nurkiewicz/Getty Images
2. Steve Metcalf - 7
Good performance at full back, doing everything that was required to hold Carlisle off for as long as possible. Made a key header off the line to deny the Blues a goal. Photo: Steve Robards
3. Jack Brivio - 7
Sturdy in the heart of the defence and his partnership with Miles kept Carlisle’s dangerous attackers quiet for 70 minutes. Photo: Derek Martin Photography and Art
4. Will Miles - 8
Solid as rock at the back. Dealt with pretty much everything the Carlisle attack attempted to throw his way. Photo: Lyn Phillips & Trev Staff