The Cumbrians spent much of the second half defending their lead, although Robson almost had a second in the six minutes of added time when he hit the crossbar with goalkeeper Joe Murphy struggling to get back in position.
The hosts' first attack saw them take the lead as Jon-Paul McGovern's cross from wide on the right was met by Robson and his header from eight yards looped past Murphy into the top right-hand corner - the winger's seventh goal of the season.
John Fleck missed a clear chance to put the Sky Blues level when he fired across the face of goal from well inside the area, then Franck Moussa was unlucky to see his deflected 25-yard strike beat Mark Gillespie but bounce back off a post.
However, Carlisle had their share of attacking in a lively first half and it took an instinctive save from Murphy to deny Brad Potts when the midfielder was put through one-on-one by an adroit Liam Noble pass.
The home team were down to nine men for a time after the break, with debutant Sean O'Hanlon and Chris Chantler off the field having treatment for head cuts, and they were forced to defend ever deeper as Coventry applied late pressure.
Leon Clarke had their best chance in the 80th minute when Blair Adams picked him out at the far post, but the unmarked striker got in a tangle with his header which he put wide from inside the six-yard box.