Noble's first goal came totally against the run of play after Rovers had forced the pace in the first half hour without really looking like scoring.
Kyle Bennett hit a fierce volley just wide in the 15th minute and Andy Griffin had a scorching 25-yard strike parried by Mark Gillespie 10 minutes later.
Carlisle had not posed a threat but they went ahead in the 33rd minute when a 25-yard free-kick from Noble took a wicked deflection off the defensive wall and looped over goalkeeper Gary Woods into the net.
Rovers came out for the second half with a renewed sense of urgency and determination, with Bennett hitting an angled shot over the crossbar in the 53rd minute.
Chris Brown had a powerful header brilliantly tipped onto the bar by Gillespie before John Lundstram saw a low 20-yard effort saved a minute later.
Brown was out of luck again when he fired against the woodwork in the 67th minute before Carlisle reasserted themselves in the closing stages.
James Berrett had a shot cleared off the line by Jamie McCombe and Noble clinched the points four minutes later when he lashed home a terrific 20-yard shot following a quick-fire break-away.