Goals from Johnny Mullins and Scott Davies gave the underdogs a sensational 2-0 lead at the break.
United adjusted to an awful playing surface faster than their hosts, but Charlton got back in the game when Michael Morrison headed home at a corner eight minutes after the break.
A flash of quality seven minutes from time earned the Addicks a third-round replay at the Kassam Stadium next Tuesday, when Yann Kermorgant volleyed home.
Wilder took heart from his side’s gutsy display, but the mood in the dressing room afterwards was bitter-sweet.
The United manager said: “There are a lot of disappointed, tired bodies in there.
“I thought we adapted well to the conditions in the first half and got our noses in front.
“We’re a bit disappointed with the manner of their first goal, which gave them en-ough time to really put us on the back foot.”
He added: “Charlton are going to have to come and have another go at us again and we’ll be up for the battle.”
The game had been postponed twice due to a waterlogged pitch and despite more than a week of intense effort by the ground staff, The Valley was only just playable.
Wilder felt the conditions suited Sky Bet League Two United more than Championship Charlton, but praised the way a changed line-up dealt with the challenge.
He said: “I’m not daft. It was a leveller for us in terms of the quality that Charlton would usually produce, as they did at the start of the season when they played us.
“We managed to make a tie of it, which we wanted to do.”
Davies, who scored one and set up the opener on a rare start, described the playing surface as “atrocious”, and if anything, was more deflated than Wilder.
He said: “If someone had said to us before the game we would get a 2-2 draw, I think we would’ve taken it.
“But with the circumstances we had at half-time, it’s very disappointing and feels almost like a defeat.”