Coventry boss Mark Robins lamented a below-par showing as his side's record run of six away wins in a row was ended by Carlisle.
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."