Carlisle were the dominant side in the first half at Brunton Park when their midfield took a stranglehold on the game without creating too many clear-cut chances.
Stanley's best effort of the opening period came when skipper Luke Joyce, the former Carlisle midfielder, tried a snapshot from the edge of the box which home goalkeeper Adam Collin needed to move smartly to his left to save.
Carlisle continued to force the pace and Stanley keeper Ian Dunbavin kept out James Berrett's near-post effort before Robson gave the Cumbrians a 39th-minute lead. Berrett did well to keep the ball in play on the right and combined with Lee Miller to find Robson just inside the area and the winger hit an unerring left-foot shot past Dunbavin into the top left-hand corner of the net.
Carlisle went off the boil after the break and Accrington striker Aidan Chippendale deserved better luck with a trio of excellent efforts, drilling one shot dangerously across the face of goal and twice forcing Collin to spring into action with fingertip saves.
The home side should have put the game to bed late on when both Robson and substitute David Symington got through the defence only to fail to convert clear-cut openings.