A game of few chances ended goalless and with no extra-time, the Cumbrians held their nerve from the spot to progress.
Liam Noble continued Carlisle's 100 per cent success rate and handed them the advantage after Ryan Williams had blazed his spot-kick over.
Kavanagh's men sealed their place in the Northern Section quarter-finals when Morecambe midfielder Andy Fleming hit the underside of the crossbar.
"It was the win we were after because it extended our unbeaten run to five games and that was really important," Kavanagh said.
"I have been taking my Pro Licence this week and we have been talking about the importance of momentum in football and that is what we have been getting here over the past few weeks.
"The lads showed great composure because there is pressure in any penalty shoot-out and we handled it really well. It is important that the club enjoys a good cup run not just for the finances but to keep our run going and we achieved that and I'm delighted for everyone.'