Two extra time goals from hat-trick man Tim Hicks took Carmarthen through to an FAW Premier Cup semi-final against Llanelli at Stebonheath Park (26 Feb).
John Toner had put The New Saints in front, but indiscipline saw red cards go to Phil Baker and Mike Taylor.
In the Loosemore's Challenge Cup semi-final first legs, Aberystwyth won 2-1 at home to Llanelli and Bangor held Rhyl 1-1 at Belle Vue.
The second legs will be played on 12 February (Bangor) and 13th (Llanelli).
Lee Hunt gave Rhyl the lead in the first leg before Mike Walsh's 83rd-minute equaliser.
At Park Avenue, Tiago Da Silva put Aberystwyth in front in the first half.
Mark Jones equalised for the Reds after the break, before Andy Evans notched a late winner for the home side.