On-loan Birmingham striker Jake Jervis scored twice for Carlisle, but each time the Robins hit back, Adam Rooney and Paul Benson keeping them on level terms by the break.
It was an error by Swindon skipper Alan McCormack which led to Carlisle taking a 14th-minute lead. His clearance bounced back into the danger area where Jervis seized on the chance to steer the ball past Robins goalkeeper Wesley Foderingham from 18 yards.
It all happened in a four-minute spell in the first half as first Rooney struck with a shot from outside the area in the 29th minute then, within 60 seconds, Jervis restored Carlisle's advantage with a blistering shot off the underside of the crossbar after Danny Cadamarteri's lay off.
The home side's lead was to last no more than three minute before Benson found room to burst into the box and drill a low shot past Carlisle goalkeeper Mark Gillespie from 10 yards.
Swindon were much the better side for a large portion of the second half and twice Gillespie came to Carlisle's rescue, thwarting Andy Williams then going down to save at the feet of substitute James Collins. The home goalkeeper excelled with an even better stop from Gary Roberts with just a couple of minutes left.