Carlisle opened the scoring after 33 minutes when Scott Dobie squeezed a pass through to Clayton, with the ball taking a deflection off Lubomir Michalik and falling perfectly into the stride of the midfield man, who coolly slid his shot beyond Casper Ankergren from just inside the area.
Leeds came out firing on all cylinders at the start of the second half and were level within two minutes of the restart. Adam Collin produced a diving save to deny Max Gradel but Snodgrass was on hand to nod the ball over the goalkeeper from close range.
The visitors were the better side for long periods in the second half but it was Carlisle who eased the pressure after 72 minutes when Hurst got the better of Michalik and turned cleverly inside the area before hammering a shot past Ankergren.
Simon Grayson's men refused to give up and Crowe lashed home a left-wing cross to keep the tie in the balance before Grella forced home Gary McSheffrey's left-wing cross to take the game to penalties.