Wexford looked in trouble when substitute defender Conor Goff received a second yellow card in the 48th minute when his side led 0-9 to 0-8.
But Byrne’s goal three minutes later was crucial in such a tight game and Wexford turned the screw for the remainder of the game.
The opening half was a tight affair, going point for point until Marty Kavanagh gave Carlow a 0-6 to 0-5 half-time lead with a pointed free in the third minute of injury time to give great hope to the home support in the 3,500 crowd.
Scorers for Wexford: I Byrne 1-5 (0-3f), G Moore 0-3, (2sl), R Clarke 0-2 (1f), B Carton, N Murphy 0-1 each.
Scorers for Carlow: M Kavanagh 0-5 (5f), S Murphy 0-2, J Doyle, D Byrne (sl) 0-1 each.
WEXFORD: B Murphy; G Foley, S O’Gorman, E Conroy; A Kenny, D O’Leary, C Heffernan; C O’Leary, J Hobbs; G Moore, L Chin, I Byrne; J Guiney, B Carton, R Clarke. Subs: C Goff for Hobbs (40); N Murphy for C O’Leary (52).
CARLOW: B Treacy; D Galway, P Doyle, D Wall; D Byrne, J Corcoran, C Wall; S Murphy, P Coady; J M Nolan, D Dalton, M Kavanagh; D Roberts, J Doyle, G Bennett. Subs: P Shaw for Bennett (36); M Doyle for Shaw (60).
Referee: G Quilty (Kilkenny).