Trailing 2-0, Carlisle staged a brave rally late in the match against a team two divisions above them in the league.
Substitute Craig Farrell pulled goal back a minute from the end to set up a fascinating climax to include injury time as Bournemouth held on to the £24,000 prize.
Bournemouth always held the edge in attack in a competitive game which exploded near the end with two bookings for Carlisle defenders Brian Shelley and Simon Grand.
Bournemouth looked sharper in the first half and Alan Connell struck a post with a shot on the turn and Derek Holmes headed straight at goalkeeper Matt Glennon with the whole goal to aim at.
Brazilian Magno Vieira twice went close for Carlisle before Bournemouth took the lead in the 48th minute.
John Spicer, on loan from Arsenal, crossed from the left and Derek Holmes blasted home a shot from 12 yards after his first effort had been blocked.
Carlisle looked doomed when Bournemouth scored again in the 64th minute with a goal of classic simplicity.
Wade Elliott who was a constant problem for Carlisle on the right wing, crossed to the far post where Holmes headed back across goal for Connell to touch in from close range.
But Carlisle were not beaten. Neil Moss was forced to go full length in the 83rd minute to keep out of fierce shot from Karl Hawley but even then Carlisle's resistance was not overcome.
Moss needed to race to the edge of his area to deny Hawley again with his legs and with a minute remaining Carlisle snatched what proved to be a consolation.
Substitute Craig Farrell bent a superb right-foot shot around Moss and into the corner of the net to set up a fierce finale.
As Bournemouth attempted to run down time, Carlisle defenders Brian Shelley and Simon Grand were booked as Bournemouth held on to their hard-earned place in the third round.