Wes Morgan, Chris Wood and Ignasi Miquel scored three goals inside 13 minutes either side of half-time to turn the match on its head as Hugo Rodallega had opened the scoring in the 18th minute.
There could be no argument it was deserved but Fulham eventually stepped up a gear and Rodallega grabbed his second before Giorgios Karagounis smashed home a free-kick with five minutes to go.
However, Lloyd Dyer had the final say in the 89th minute to put Leicester into the quarter-finals.
"It was a good game, two sides trying to win it. I am probably a bit more relaxed than Martin but it's been a good night for us," he said.
"You can look at the goals they scored and talk about the possible difference between the Premier League and Championship when it comes to taking chances.
"We have a competitive squad, a squad of players who aspire to be in the Premier League - as any number of Championship clubs - so let's not get ahead of ourselves.
"It could have gone either way but we've come out on top.
"I'm not trying to be clever and we mustn't read too much into the result.
"But, having said that, our players deserve an awful lot of credit for coming out on top against a side who are not having a great time in the Premier League."