The newly promoted Cumbrians recovered from conceding an early goal to take control of the game and claim three strikes of high quality before a late penalty set up a tense finish.
The home side, who were unbeaten until last week's defeat at Wrexham, grabbed the lead after 19 minutes when midfielder JJ Melligan collected the ball in midfield and speared a 25-yard shot beyond goalkeeper Anthony Williams and just inside his left-hand post.
The lead lasted only five minutes, however. Cheltenham midfielder John Finnigan dwelled on the ball for too long and was robbed on the edge of his own box by Chris Billy. Striker Karl Hawley was onto it in a flash and guided a low shot beyond goalkeeper Shane Higgs.
Both sides had a number of half-chances either side of half-time but the turning point arrived on 52 minutes. Carlisle left-back Zigor Aranalde found Hawley on the edge of the area and the former Walsall man turned his marker before shooting past Higgs.
Aranalde seemed to have put the result beyond doubt 78 minutes when he scored a spectacular third for the visitors. Chris Lumsdon's corner was headed out by Ashley Vincent but Aranalde thumped a spectacular volley back from the edge of the box and into the top right-and corner of Higgs' net.
Cheltenham had struggled for any sort of fluency going forward, although they were handed a late lifeline when Carlisle captain Kevin Gray jumped all over Jamie Victory in the penalty area. Referee Jarnail Singh pointed immediately to the spot and Grant McCann despatched the penalty.
Cheltenham threw players forward in the closing minutes but Carlisle were rarely troubled and can now boast an impressive record of three wins and two draws on the road.