Connah's Quay suffered the heartbreak of penalty shoot-out exit after also losing key defender Chris Williams early in the game with a suspected fractured cheekbone.
Williams, who spent part of last week at Millwall and was due to trial with Tranmere next Monday, was stretchered-off and taken to hospital after just 24 minutes.
But despite having to reorganise their defence, and also losing striker John McAllister at half-time, the Nomads gave a good account of themselves. Jamie Petrie fired over in an early attack, but Carmarthen also looked threatening with Jamal Easter only denied by a goal-line clearance and then firing wide when well-placed.
Stuart Cook should have put Nomads ahead shortly after the break when somehow missing from close range with only keeper Neil Thomas to beat but, with the hosts enjoying their best spell of the match, the Old Gold took the lead against the run of play when Nicky Palmer fired home from just inside the box.
Stewart White levelled 12 minutes later shortly after coming off the bench, but it was keeper Pritchard who took the game into extra-time with two superb saves.
With honours still even at the end of 120 minutes, Pritchard then saved Town's second penalty, but Thomas saved three at the other end to ensure the West Walians a long but happy journey home.