The League One side won a thrilling penalty shoot-out, while Leeds United came from behind to beat Chesterfield and Notts County got the better of fourth tier Fleetwood Town
David Symington was the hero for the League One side as he fired down the middle to hand Carlisle a 4-3 victory on penalties after an exhilarating 3-3 draw following extra time.
Alex Salmon fired the first penalty wide to hand Blackburn the advantage but Chris Taylor and Josh Morris missed the last two spot kicks as the 2002 League Cup winners exited the competition.
Earlier, David Amoo had fired the home side ahead in the 14th minute at Brunton Park as Greg Abbott's men led at the break, but two goals in five minutes from Cairney and Taylor in the second half put the Championship club ahead.
Carlisle refused to lie down, however, and Lewis Guy's equaliser ensured the game would head for extra time.
Cairney provided the cross for Alan Judge to put Blackburn ahead during the first half of extra-time before he received his marching orders for a second bookable offence, and the match went to penalties when Amoo notched his second with six minutes remaining.
That led to the dramatic shootout where it looked as though Rovers would go through after they were leading 3-2 following Salmon's early miss, but Carlisle would once again show the fighting spirit they had throughout the 120 minutes to come back and earn their spot in round two.
Rovers coach Gary Bowyer rued the red card shown to Cairney and believes it was a harsh decision.
"From a neutral's point of view it was an entertaining cup-tie, it has had everything," he said. "A missed penalty, a sending off, end-to-end, but in terms of our performance I thought we were good with the ball and we should have out of sight.
"We find ourselves a goal down you saw our character, we carried on playing, we got back in the game and got the equaliser. Obviously the game changes a little bit on the sending off which is harsh, the referee couldn't wait to get his red card out.
"We've been on the wrong end of a couple of decisions in the two games and I'm sure it will even itself out over the season. We've just got to re-group, dust ourselves down and get on with trying to win on Saturday."
Elsewhere, goals from Michael Brown and Dominic Poleon saw Leeds United come from behind to defeat Chesterfield 2-1 at Elland Road.
Eoin Doyle had put the away side ahead in the 17th minute but they were ultimately unable to knock out their Championship opponents.
In the night's other game, League One side Notts County defeated Fleetwood Town 3-2 to progress to round two.