The sides were locked at 0-11 each with just five minutes remaining when Brendan Halpin and Derek Butler put them two clear.
Kildare’s Padraig Fogarty was fouled on his way to goal and successfully kicked the resulting free but there was to be no leveller.
Michael Regan’s frees had given Wexford a 0-5 to 0-3 interval lead.
A late start, due to a power failure, helped Kildare settle and they went ahead with a Padraig Fogarty point. But in a tight and tense finish, it was Wexford who prevailed.
Scorers for Wexford: K O’Grady (4f), M O’Regan (3f, 1 45) 0-5 each, D Butler, L Óg McGovery, B Halpin 0-1 each.
Scorers for Kildare: P Fogarty 0-6 (1f), J McAndrew 0-3 (2f), F Dowling 0-2, S Hurley 0-1.
WEXFORD: R Farrell; J Breen, S Gaule, C Carty; D Butler, J Leacy, L Óg McGovern; M O’Hanlon, M O’Regan; D Murphy, E Kent, P Kelly; K O’Grady, K Kehoe, B Halpin.
Sub: M Furlong for Breen (52).
KILDARE: C Heneghan; D Holton, S Campbell, M Kelly; J Comerford, A Karey, C Plunkett; K Feeley, R Kelly; P Fogarty, E Heavey, S Hurley; F Dowling, P O’Neill, J McAndrew.
Subs: W Burke for Plunkett (46), D Egan for O’Neill (52).
Referee: J Curley (Meath).