The Model county had built an early 1-2 to nil lead and stayed ahead all the way until Meath substitutes Adam Ducie and Senan Duffy looked to have snatched victory from their grasp with points in additional time.
Wexford had one last chance, and there was pandemonium in their dug-out after Kian Steadmond teed up Funge to cut in from the left and blast in the winning goal in the 64th minute.
The outstanding midfielder had netted already in the 11th and 17th minutes after soloing from the Meath 65-metre line before producing two clinical finishes.
His first goal followed a pointed free from Odhran Morris, and the same player added a second to leave Wexford clear by 1-2 to nil.
Meath got off the mark with a Cormac Smith free in the 13th minute, and Evan Kelly, Cormac Nugent (free) and Smith from play made it a one-point game before Funge’s second superb goal.
Wexford held that 2-2 to 0-4 lead to half-time, and they were still ahead by 2-6 to 0-6 entering the last quarter before the accuracy of Cormac Nugent in particular saw Meath gradually chip away at the deficit.
Four points on the trot from Pat Crawley, Nugent, Ducie and Duffy looked like breaking Wexford hearts at the death, but the inspirational Funge had other ideas.
Scorers – Wexford: T Funge 3-0, O Morris 0-5 (4f), A O’Neill, J Hughes 0-1 each. Meath: C Nugent 0-5 (3f), P Crawley, E Kelly, S Duffy, C Smith (1f) 0-2 each, A Ducie 0-1.
Meath: C Murphy; S McQuillan, C Yore, T Sheridan; C Tobin, O Meade, C Diggins; C Barron, E McCrudden; C Nugent, C Smith, C Sheridan; E Kelly, J Doyle, P Crawley. Subs: S Duffy for Sheridan (HT), C McCarthy for Diggins (43), H Healy for Smith (44), A Ducie for Meade (50).
Wexford: J Gallagher; K Steadmond, L Kenny, S Fitzharris; R Cooper, E Hughes, F Kavanagh; T Funge, J Rothwell; J Hughes, O Morris, D Forde; I Noctor, D Rossiter, A O’Neill. Subs: A Kelly for O’Neill (47), M Beary for J Hughes (55), A Carton for Rothwell (55), C Power for Forde (58).
Ref: N Ward (Westmeath).