Leicester manager Ranieri made 10 changes to the side that lost 2-0 at Swansea City last week -- a fifth straight league defeat -- but even when Millwall had Jake Cooper sent off after 52 minutes the English champions failed to take advantage.
Shaun Cummings then slotted a 90th-minute winner for the League One hosts who claimed their third top-flight scalp after victories in the previous two rounds against Bournemouth and Watford.
"I am very disappointed. We played well in the first half, so-so at the beginning of the second until it went to 10 versus 11," the Italian told the BBC.
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"After then they played better than us, with more desire and heart and deserved to win. We have to show desire. I gave a great opportunity to my players.
"We can learn. When a team from League One beats the champions we say, why and have to react as soon as possible. We are better than Millwall, but Millwall deserved to win."
Leicester failed to score for the seventh time in their last 10 games and the news got worse as it was confirmed that January signing Molla Wague, handed his debut, suffered a dislocated shoulder during the fifth-round tie at the New Den.
With a Champions League last-16 clash away to Sevilla on Wednesday, followed by a home Premier League match against Liverpool, the games are coming thick and fast for Leicester, who face a desperate battle to avoid relegation a year after stunning world football by winning the title.
Ranieri said that was why he made so many changes.
"When you play three matches in nine days, you need to give the opportunity," he said. "If you play well, you don't change too much, but when you don't play well, you lose a match, you have to give the opportunity to the other players."
Millwall, beaten finalists in 2004 when they were in the second tier, are threatened with having to move from their south London home because of a dispute with the local council who want to sell adjacent land to developers.
But all that was forgotten on Saturday as the fans were given a welcome boost.
"When we went to 10 men it galvanised us. There was more pressure on Leicester," manager Neil Harris said.
"I made attacking changes to win the game because neither of us wanted a replay. I thought the atmosphere was electric. The noise was phenomenal. These are special days for us."
(Reporting by Martyn Herman,; Editing by Neville Dalton)
