For 120 minutes both sides served up a cup classic with six goals, a missed penalty and a dismissal.
David Amoo's early goal for Carlisle was cancelled out by a Liam Noble own goal before Chris Taylor gave Rovers the lead again - only for Lewis Guy to level. Substitute Alan Judge then blasted Rovers back in front in extra time only for Amoo to make it 3-3 and take the game to penalties.
Teenager David Symington then lashed home the decisive spot-kick in the shoot-out after Taylor and Josh Morris had both missed for Rovers.
There was nothing complicated about Amoo's goal in the 14th minute. Josh Gillies fired a corner across the penalty area and the Carlisle winger took advantage of some poor defending to direct a downward header past Simon Eastwood from six yards.
There was, however, a big slice of fortune about Rovers' 52nd-minute equaliser. Noble stuck out a foot to block Tom Cairney's 20-yard drive, but he only succeeded in producing a huge deflection to send the ball away from his own goalkeeper Mark Gillespie's grasp and into the goal by the right-hand post.
Two minutes later Blackburn's Josh King won a penalty after Danny Livesey had challenged clumsily inside the area, only for David Dunn to see Gillespie produce a fine diving save.
There was no holding Rovers back when Taylor grabbed his first goal for Rovers in the 56th minute, rising highest at the far post to nod home Cairney's corner.
Carlisle hit back immediately, though, Guy forcing a header home from close range in the 62nd minute after Mark Beck had done all the legwork on the right wing.
Judge - introduced as a substitute for extra time - quickly made his mark, rifling home right-footed from 12 yards after 95 minutes of play.
Rovers would finish the game with 10 men after Cairney was cautioned for a foul on Beck before being red carded.
There was still more drama to come in the second period of extra time as Amoo cut in from the left to hammer a shot into the bottom right-hand corner and draw Carlisle level and into a dramatic penalty showdown.