The Blues ran out 2-1 winners at The Shay on Tuesday night to book their place in the semi-finals away at Solihull Moors on Sunday.
A header from Danny Rowe and a brilliant strike from Jeff King against his former club secured the victory for Paul Cook’s men.
"I think we've had all the ball, dominated the game, we've limited them to half-chances and the two goals we've conceded are disappointing, goals we don't normally concede, which is utterly disappointing," Wild told the Halifax Courier.
"But for all our endeavour, our qualities, those moments have let us down tonight.
"I thought we were excellent second-half, I thought we moved the ball well, we just didn't create enough chances.
"But we've got a lot of players who've played a lot of games, who've never played that amount of games before, and I think in the last two weeks it's caught up with us.
"I always thought we were going to get something, I never felt like we were never going to get something.
"We were so on top, I just never felt like we were going to lose.
"But I think unfortunately, our quality in those moments that matter has cost us this evening."
The hosts felt that Liam Mandeville should have received a straight red card for a tackle on Jack Senior in the first-half but referee Scott Tallis opted for a yellow instead.
“I think it actually took a red card incident to liven the game up,” Wild said.
"I thought it was a good game after that little tussle, but we all know the referee has made the wrong decision. There's three of them within five yards and they can't make the right decision, it's just baffling.”
A crowd of more than 5,600 watched the entertaining clash at The Shay.
"Great atmosphere, I'm just gutted we couldn't do it for them tonight,” Wild added.
"But unfortunately I think our race was run. And I think if we'd gone on to the weekend that would have exposed us even more.
"Unfortunately, we are where we are."