For the second time in three days Jeff Hendrick proved to be Derby's man on the spot, grabbing the second-half goal which broke Carlisle's brave resistance and secured a 2-0 win in the Capital One Cup first-round tie at Brunton Park.
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Hendrick, who netted the winner in Derby's Sky Bet Championship opener on Saturday, was unmarked at the far post when he headed in Will Hughes' cross, and Derby secured victory in added time when Chris Martin sidefooted a right-wing cross home from six yards.
Teenage forward Kyle Dempsey almost marked his full debut for Carlisle with a goal when he fired in a shot from the left edge of the area which was deflected over Lee Grant's crossbar for a corner.
The League 2 side made a bright start, but Derby dominated as the first half went on and Mark Gillespie did well to deny Jamie Ward who screwed his follow-up shot wide of the far post after the goalkeeper pushed out his initial header.
Gillespie saved brilliantly from Martin's header but there was nothing he could do minutes later when Martin lashed in a shot which had the goalkeeper beaten but glanced off the bar.
Carlisle almost shocked their Championship opponents with a 25-yard free-kick by David Symington which rattled the crossbar. Grant got his fingertips to the shot to make a crucial deflection.
Derby finally broke the deadlock in the 62nd minute when Hendrick drifted in at the far post to nod home Hughes' right-wing cross from two yards out, and Martin added the second in the final stages.