It needed a typically predatory finish from Cahill, who reached his half century of goals in his 192nd appearance for the Toffees, eight minutes from full-time to shatter Carlisle's resistance before Leighton Baines put a flattering gloss on the score with a late penalty.
Carlisle were never overawed by their more illustrious opponents and, backed by 6,000 fans who had braved blizzards to get to Merseyside, they were unfortunate not to earn a money-spinning replay.
But despite that setback Carlisle were level within six minutes as Clayton made amends for his mistake.
Vaughan thought he had restored Everton's lead when he turned in Cahill's shot but his strike was ruled out for offside.
Young striker Vaughan, only just back from a long-term injury, went close again with a diving header from Baines' cross.
Carlisle were more than holding their own as Everton struggled to impose themselves and the visitors were agonisingly close to taking the lead when captain Danny Livesey rattled the bar from 12 yards.
Just as Everton seemed to have run out of ideas, they finally took control in the 82nd minute.
Cahill saw one shot blocked by Ian Harte, but regained possession to fire past Collin.
An injury-time penalty from Baines, after Collin had brought down substitute Keiran Agard, secured Everton's place in the fourth round.
Source: AFP Global Edition