Jayden Bogle's second-half goal secured the win - Wilder's 100th as United manager, and their first in any competition since they beat Chelsea 3-0 in the Premier League on 11 July.
"This was a big day for us - because of what's happened, it's been a tough week," Wilder said.
"The consequences of not getting a result today would have really impacted on the group. The players have taken a lot.
"Everybody in the country would have sniffed this. Every expert, pundit and supporter wouldn't have been surprised if we had been turned over today. But there was a good attitude.
"It was always going to be a tough cup tie, away from home and not being in form at all. It was against a team with nothing to lose and we had everything to lose today."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!