Max Power's superb 25-yard strike after nine minutes gave the Black Cats a 1-0 win at Bramall Lane and it was the visitors who came closest to scoring the game's second goal when Luke O'Nien's shot struck a post in the second half.
The opening seven minutes aside, the Blades offered very little and few could deny Sunderland their victory.
The hosts made 10 changes. Only Callum Robinson survived from the team who started the Premier League win at Everton on Saturday, and Wilder insisted his fringe players had missed an opportunity.
"We didn't do enough and we didn't deserve to go through, we were second best," said Wilder.
"There would have been a time when I'd have had them in the dressing room dissecting that until 11:30pm but we've got the European champions coming here at Saturday lunchtime and I'm not going to waste my time on it.
"It was an opportunity for some players to knock on my door and ask why they aren't in the team, well I don't think there will be any knocks on my door, apart from maybe Mo Besic. And I thought the two strikers who came on did okay.
"I made 10 changes and Sunderland made eight changes and their changes were better than mine. They were pretty comfortable in the end and saw the game out comfortably.
"Some of my players had an opportunity and they have not taken it, that's their problem. We move on."
Sunderland, finalists in this competition as recently as 2014, will travel to League One rivals Oxford in the next round.
Black Cats boss Jack Ross said after the win: "There's a togetherness in the squad and it is a really positive night for us.
"Nothing should be taken away from what we have done tonight."