Carlisle 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.
Carlisle's 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 Coventry 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 Carlisle midfielder was put through one-on-one by an adroit Liam Noble pass.
Carlisle 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 the home side were forced to defend ever deeper as the Sky Blues 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.
First-half goals from Febian Brandy and Paul Downing set Walsall on their way to a third straight win - 3-1 at Preston, who slumped to a third straight defeat.
Former Manchester United striker Brandy opened the scoring for the Saddlers in the 18th minute before Downing doubled their advantage just before half-time.
James Baxendale put the result beyond doubt after 56 minutes, with Bailey Wright's late header unable to prevent Preston's winless run at home in the league extending to nine.
Walsall went close after just five minutes with North End goalkeeper Thorsten Stuckmann denying both Jamie Paterson and Andy Butler with fingertip saves.
Preston striker Joe Garner, making his home debut, was denied a 16th-minute penalty by referee Carl Boyeson despite being upended running on to Stuart Beavon's low cross.
Walsall took full advantage moments later when Brandy turned in Will Grigg's cross before Downing added a second in the 38th minute with a header from an Adam Chambers delivery.
Baxendale completed the win after the break when Luke Foster's loose touch allowed Grigg to pick him out, before Wright's stoppage-time consolation for Preston.