After the game ended 0-0 in 90 minutes, penalty misses from Ryan Williams and Andy Fleming proved costly as Carlisle scored all their spot-kicks to secure victory.
Liam Noble gave the Cumbrians the advantage after Williams had blazed his spot-kick over and when Fleming hit the underside of the crossbar the Shrimps were out.
The game produced few chances in the 90 minutes.
Carlisle went closest to a goal after 41 minutes when Robson let fly with a free-kick from 25 yards out that was superbly saved by Morecambe keeper Andreas Arestidou diving to his right.
Morecambe looked far more threatening in the second half and Carlisle keeper Mark Gillespie was forced into the best save of the game after 73 minutes when Kevin Ellison's header deflected off visiting defender Paul Thirlwell and was heading for the top left-hand corner until Gillespie saved superbly to his left.