The Premier League club were trailing 2-0 with 12 minutes to go after goals from Patrick Bamford and Izale McLeod put MK Dons on the brink of an upset.
However, Jozy Altidore pulled one back and then substitute Connor Wickham's brace and Adam Johnson's goal in the closing three minutes sent Sunderland through.
Di Canio said: "It was very bad, but I know also it is easy to play badly in this game. Unfortunately, it happens even to the top clubs who play with the top players in this league.
"It's something that have thought for many years, why do top teams in the cup in England have to go out easily against League One and League Two sides? I don't understand.
"But I understand [now]. I was worried, and I was right, but I took something really good from the last 15 minutes, the ambition, the desire, the pride and the goals that we scored in style.
"I am very, very happy for 15 minutes; I am very, very angry for 75, so the balance is not very good."
The Sunderland manager praised the impact of Wickham, who has struggled to make a sustained impression since his move from Ipswich Town.
Di Canio said: "He turned the game. He has improved a lot in the last few weeks, I have seen a very good improvement, not because he scored two goals tonight, but his body language and desire.
"A few days ago, we had a chat for 45 minutes together to help him because he is still only young. He is a big fella, but still young, so the fact that he wanted to hear some words help him and give him more confidence.
"This is why you saw him today for 40 minutes have the quality of a player for a Premier League club."
MK Dons boss Karl Robinson was delighted with the quality of his side's performance in a competition which he admits is a secondary consideration to the League One campaign.
He said: "I have just gone in and had a bit of an argument with some of the players because I went in and said 'Well done' and they are saying - well, I can't repeat what they said, but it was along the lines of being disappointed, saying, 'That defending is not good enough, we have got to learn from it'.
"They are disappointed not winning here tonight, so that in itself is a tremendous testament to them.
"I have got young kids - Dele Alli, let's be honest, that kid has just turned 17 and I think you will be writing an awful lot of lines about this young boy many, many times because he's as good as I've seen.
"You have all seen tonight that he's destined for going in one direction. He's got all the attributes. He's in there fuming that we have been beaten at 17 years of age."