Bournemouth extended Howe's unbeaten run since his return to the club to 11 games as they saw off a late recovery by Carlisle to complete a double, having beaten the Cumbrians 4-2 in a league visit a month earlier.
First-half goals from Wes Fogden and Eunan O'Kane set up the win for Bournemouth on Saturday and although Mark Beck's header gave Carlisle some hope, the outstanding Marc Pugh clinched it for the south-coast men in time added on.
Howe said: "It's important for us to get in the draw for our supporters. All credit to the players to come to a difficult place and respond brilliantly to the challenge.
"We controlled the game and played well from back to front. I don't remember Carlisle having that many chances and from our point of view the only negative was the goal because otherwise it was a really good performance."
The Cherries boss added: "There was always a chance it would be dicey towards the end, but we remained in full control and saw it out professionally with the third goal the icing on the cake. Two or three of them took their chance here. We'll try and beat whoever we draw in the next round and I just want us to go as far as we can."