Carlisle gave as good as they got but, in the end, it was the heartbreak of a mammoth penalty shoot-out which ended their EFL Cup dreams.
The home side started in bright form and Gillespie had to be careful as Bent bore down on a back pass from Joyce. The keeper kept his cool and hooked the ball clear under pressure.
The Rams had a real chance on two minutes when Blackman chased a clip over the top from midfield. It was a tight angle and his first attempt was blocked by the alert Carlisle stopper. Blackman was up like a flash but he could only stab it into the side of the net.
A neat run from Lowe earned the hosts a corner and there was a real moment of worry as a downward header spun behind off Joyce's hand. The referee ignored the appeals and waved play on.
More good work from Derby took them down the right flank and Camara did well to step through one challenge before cutting into the box. He went for the shot but it rose over the angle of post and bar.
The Carlisle travelling contingent were in good voice on 12 minutes when they watched their team keep the ball for a long run of consecutive passes. The move was brought to an end when Carson bent low to pluck a Grainger cross off the grass.
A whipped delivery from Olsson had to be cleared when he raced wide to send it in as Derby rallied, and Lowe dropped a shoulder to beat Adams before he also fired the ball into the mix. Gillespie did well to watch it all the way before pulling it into his chest.
Striker Darren Bent twice got in the way of his own men as he first blocked a shot from Camara in its tracks, and then deflected the follow-up from Butterfield wide of the target.
The Cumbrians had a huge appeal for a penalty turned down on 22 minutes when Adams worked space for a cross on the left wing. Grainger got his head to it and the flight appeared to be altered by an outstretched hand. The referee thought about it then pointed to the corner flag.
A quickly taken set piece routine followed and Ellis was disappointed to see his back post header fly straight at Carson. The Rams broke quickly from the throw out and Blackman used pace to get to the edge of the area. Gillespie had to go full length to parry a thumping shot which was angled at the bottom corner.
Bradley Johnson was next to lose his bearings as he crunched one wide from 20-yards, and Gillespie had to be off his line at pace when a through ball invited Bent to run on. The big stopper reached out to fist it away.
Carlisle were unlucky again just before the break when Devitt controlled a high ball forward with skill. He took two touches and put Miller on the overlap, but the offside flag accompanied a fine Carson save.
Jamie Devitt turned defence into attack when he picked the ball up deep inside his own half and in acres of space. He remained unchallenged as he stepped within shooting distance but his effort lacked power and it curled harmlessly behind.
A drop of the shoulder from Camara and a brisk poke over the bar signalled the start of the second half as Derby came sharply out of the blocks. Camara was prepared to have a go again moments later when a half clearance dropped out of the sky. His volley was accurate but easy pickings for Gillespie.
The Cumbrians had half a sight for themselves at the other end when Adams dropped deep to take possession of the ball. His shot was fierce but Keogh was on hand to make a block.
Abdoul Camara was suddenly in a lively mood and he beat the floor in frustration as he went past two men only to shoot high.
The chance of the game thus far came for the visitors when Danny Grainger arrived late on a left flank run. He spotted Devitt and the pass was perfect as it skipped up off the slick surface. The nippy wide man got his head over it but it wasn't enough to prevent it from going just an inch too high.
It proved to be a costly miss as the Rams took the lead with the very next attack. Blackman thought he'd done enough with a wonderful strike which brought a world class save from Gillespie as he somehow got there to tip it onto the woodwork. He could do nothing as Bent latched onto the rebound and swept it into the waiting net.
Manager Keith Curle reacted immediately by throwing McKee, Ibehre and Wyke into the mix, but Blackman almost struck from a long ball forward which eventually squirmed into his path. He caught it well but it bounced wide of the upright.
Blackman and Ince had wild efforts which did little to trouble anything other than the fans in the seats behind the goal, and Will Hughes produced more of the same when he curled an Ince cut back into the ozone layer.
Joe McKee put a late volley right at Carson as United stayed on the hunt and they got their reward with the last kick of the game when Jones held his position just outside the box. It dropped onto his foot and the keeper had no chance with a dipping volley which skimmed high off the bounce and smacked into the back of the net.
Tom Ince scuffed a shot on the run wide at the start of the first period of extra time and Carlisle went close at the other end when Raynes flicked a Tom Miller throw into the six yard box. Grainger tried to make it his but Keogh intervened.
And they were even closer when a Grainger free kick brought a firm header from Miller. Carson was beaten but an alert defender scurried back to hook it away.
Gillespie was in action again at the end of the first period of extra time when he threw himself across to push a blistering Ince shot behind.
Blackman struck one just wide at the start of the second half, and he lofted another over the bar as he slipped when Will Hughes played him through.
Substitute Timi Elsnik was next to go close when he edged one wide from long range, and Keogh popped a far post header into the side netting when he thought he'd nicked the tie late on.
Will Hughes somehow missed from 12 yards when he looked odds on to score, and Gillespie parried from Ince as home pressure started to mount. But United settled things down to take it to the lottery of the penalty shoot-out.
Devitt - Perfectly placed - 1-0
Ince - Sent Gillespie the wrong way - 1-1
Jones - Another great spot kick - 2-1
Blackman - Again sent the stopper the wrong way - 2-2
Joyce - Absolutely no mistake, drilled - 3-2
Butterfield - Stuttered and sent Gillespie the wrong way - 3-3
Wyke - Too close to the keeper - SAVE - 3-3
Hughes - Great save from Gillespie - SAVE - 3-3
Grainger - Drilled it low - 4-3
Keogh - Found the corner well - 4-4
McKee - No mistake, a beauty - 5-4
Elsnik - Top corner, great penalty - 5-5
Gillesphey - Squeezed it in off Carson's hands - 6-5
Baird - Sent keeper the wrong way - 6-6
Ibehre - Carson got low to make a one-handed save - 6-6
Pearce - Lofted it over the bar - 6-6
T Miller - Drilled a lovely one in - 7-6
Lowe - Gillespie touched it but the power took it in - 7-7
Raynes - Top corner, absolute corker - 8-7
Carson - Very well placed - 8-8
Devitt - Rifled it again, great finish - 9-8
Ince - He finished well for a second time - 9-9
Jones - Wrong-footed the stopper - 10-9
Blackman - Stuttered and scored - 10-10
Joyce - Swept another killer into the corner - 11-10
Butterfield - Another pause then a great strike - 11-11
Wyke - Keeper could only watch - 12-11
Hughes - A clean hit - 12-12
Grainger - Put it into the far side - 13-12
Keogh - Former-Blue scored another cracker - 13-13
McKee - Long run up but a good save - 13-13
Elsnik - Ended it in favour of the hosts - 13-14
Derby County - Bent (57)
Carlisle United - Jones (90+5)
Carlisle United - Grainger (108)
Derby County - Carson, Keogh, Bent (Hughes 66), Baird, Johnson, Pearce, Butterfield, Camara (Ince 66), Blackman, Lowe, Olsson (Elsnik 73). Subs - Grant, Martin, Hanson, Rawson.
Carlisle United - Gillespie, T Miller, Raynes, Ellis (Ibehre 64), Gillesphey, Grainger, Joyce, Devitt, Jones, Adams (McKee 55), S Miller (Wyke 65). Subs - Crocombe, Atkinson, Penn, Lambe.
Referee - Mr Darren England
Attendance - 9,860 (611 away fans)