Latching onto Andy Taylor's cross, Moore turned and fired past Ben Williams thanks to a deflection off defender Richard Keogh that looped the ball over the flat-footed keeper. The 2-1 defeat was generous on the Cumbrians and Tranmere could easily have won by a cricket score. Carlisle stuttered from the first minute to the last and boss John Ward must now be regretting the summer sale of star keeper Keiren Westwood, which has left the side in dire need of confidence.
Edrissa Sonko almost grabbed a second for Rovers inside ten minutes but Williams tipped over his smart header after Ryan Shotton's long throw had bewildered the home defence. Seconds later Sonko dragged a rasping volley just wide as Tranmere swarmed forward.
With Carlisle seemingly happy to roll over, Williams was on hand to keep out a stinging 30-yard drive from Moore before some more terrible defending allowed Rovers captain Antony Kay to head wide. Carlisle slumped even further in the second half and Moore smashed an angled drive against Williams' crossbar as Rovers looked to cement their lead.
Another Moore run and shot was only halted by a last-ditch tackle before Kay's snapshot from 18 yards was blocked by Williams. Tranmere finally made the game safe with ten minutes to play when Shotton beat the offside trap and headed past the flapping Williams from Moore's lofted cross.
It was the least the visitors deserved but, with the game deep into stoppage time, Carlisle grabbed a consolation goal when Danny Livesey's header struck the inside of the post and rolled across the line for substitute Danny Graham to tap-in.