Coventry manager Mark Robins lamented a below-par performance from his side as a second-minute goal from Matty Robson ended the Sky Blues' record run of six away wins in a row.
Robson's header from eight yards looped past Murphy into the top right-hand corner to open the scoring, and although John Fleck, Franck Moussa and Leon Clarke all had good chances for Coventry, they failed to find an equaliser.
"Rubbish" was Robins' initial response.
He added: "We didn't get the performance we expected and hoped for bearing in mind what we have done over the last two-and-a-half months. It's difficult to pinpoint anyone who had a good contribution to make.
"That certainly wasn't what I was looking for. We lost our shape and we've got to work better than we did here. It was a massively bad goal to concede and it put us on the back foot from the first minute. We made too many bad decisions, but we have to take it on the chin and move on.
"We missed some gilt-edged chances and too many had an off day. I don't blame lack of effort, but no excuses, we need to produce better than that."
Carlisle manager Greg Abbott hailed "an unbelievably strong" performance from his side.
"It was extremely tough - the best 1-0 you will see. It's fantastic to be involved in a game like that. We weren't very good at our job last week at Swindon, but we have beaten a strong Coventry side on a good run."
Abbott praised new signing Sean O'Hanlon, saying of the ex-Hibernian and MK Dons defender: "He played like a proper number five."
O'Hanlon and full-back Chris Chantler both had to receive stitches for head wounds and Abbott added: "I told the lads it was nice to see blood in the dressing room. Both players needed stitches. Chantler was sick the night before and wasn't going to play so he was magnificent."