The north London club's 36th game of the season, more than any other English top-flight team, was going to plan after Andros Townsend's early penalty but late goals by Leonardo Ulloa and Jeffrey Schlupp sealed a fifth round place for Leicester.
Tottenham, sixth in the Premier League, will try to defend a 1-0 first leg advantage in their League Cup semi-final at third-tier Sheffield United on Wednesday and resume their Europa League challenge in the last 32 against Fiorentina next month.
Spurs manager Mauricio Pochettino has rotated his squad freely and did so again with record signing Erik Lamela back from injury and Spain striker Roberto Soldado in attack.
Belgian international Moussa Dembele also started, as did Brazil midfielder Paulinho, but Tottenham were punished for failing to build on their lead as Leicester pounced late on.
"I'm very disappointed because I thought we played well," Pochettino told the Spurs website (www.tottenhamhotspur.com). "The problem was we didn't kill the game and you need to kill the game.
"We gave the opportunity to Leicester to stay alive in the game and in the last 10 minutes we lost everything we made in the first 80 minutes.
"We need to show character and we need to go to Sheffield and try to get the victory," added the Argentine.
Leicester's main priority is to stay in the top flight but manager Nigel Pearson said an FA Cup run was a useful boost for the months ahead.
"Today's result, against a Premier League side with an awful lot of quality, will hopefully be a positive influence on the rest of the season," he said.
Pearson gave a debut to new signing Andrej Kramaric and the Croatia midfielder was denied a first-half penalty.
"Today was a really good experience for him. It gave us a chance to use him in a game, which allows him to acclimatize against a Premier League side," said the Leicester boss.
"He is going to be an important player for us."
(Reporting by Martyn Herman; editing by Ken Ferris)
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