Kian Spence gave Town the lead, a goal which was quickly cancelled out by Conor Branson, while The Shaymen were indebted to keeper Sam Johnson for some vital saves as well as keeping out three Alfreton penalties in the shoot-out.
"I looked at them (Town) on Thursday and Friday and they looked goosed in training, they looked out on their feet, and I knew it was going to be tough today," Wild said.
"It's a balance between trying to give players a rest with making sure we got through to the next round.
"You chuck into the mix we're now where we are in the National League and everybody wants to beat you, I said that to the players, everybody now sees you as a scalp so you've got to match them, which I don't think we did first-half, we looked tired and sluggish.
"When we scored the goal, it was our first bright bit of play and what we'd worked on, trying to suck their full-backs out and play beyond them.
"Then it was disappointing they scored off a set-piece, I said to the players that they'd only scored one off set-pieces this year.
"Second-half's a non-entity, it just stunk of pens.
"But people keep saying to me that we always find a way to win and we've found another way of winning, which is fantastic."
Spence's goal was Halifax's only shot in the entire game.
"What's happened against every National League North side we've played is they've bedded in, made it hard for us, had a right go and made it tough," Wild said.
"It was a completely different challenge today that we haven't faced this season."
Alfreton were the better side in the game but will rue squandering some good chances.
"They probably were the better team, Billy should be dead proud of how they've played today because they've given us a right good going over," Wild said.
"But we found a way. What pleases me is we're finding all different ways of winning, whether it's dominating the football, bedding in or penalty shoot-outs."
On man-of-the-match Johnson, Wild said: "He's had to keep us in the game, but goalkeepers at every level keep you in the game.
"I say to him every week 'we'll need you at some point' and we've needed him today.
"In the shoot-out you're always dead confident he's going to save pens."
Wild made few changes to his side for the game, fielding a strong line-up.
"I want to win everything we're in, every game we play," he said.
"And when you're on a good run, players want to play.
"There were a couple who I felt needed a break but I wanted to play as strong a line-up as I could."
And the Town boss confirmed striker Matty Stenson is expected to be available for next Saturday's trip to Maidenhead after missing out today.
"The kid's got the worst luck in the world - he's now ill," Wild said.
"The kid deserves a break at some point and I hope that comes soon for him."