The Cottagers are 14th in the Premier League with 10 points, two points above the relegation zone, having won just three times this season so the result - considering Jol made nine changes - was not that unexpected.
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, Dyer had the final say in the 89th minute to put Leicester into the quarter-finals.
With Fulham's next two matches at home to Manchester United and away to Liverpool the cup defeat could not have come at a worse time on the back of a loss at Southampton.
"I don't feel pressure because we're not bottom of the league, there are six or seven other teams on similar points and lower," said the Dutchman.
"If people tell me I've got pressure then I've got pressure but for me it's no problem.
"There are a few teams on 10 and 11 points so we have to come up with a surprise result (against United or Liverpool) and if not we have to get the points against the other teams we play against.
"A season is more than two games. Of course it is not great after the Southampton game.
"We played an almost perfect second half against Palace (a 4-1 win) and we played the same team and lost against Southampton so we have to pick ourselves up.
"A few boys showed tonight they could play and I thought the spirit during the game was good - but we gave cheap goals away.
"The disappointing aspect is that without playing perfect we controlled the first half and then there was the craziness of them scoring two goals out of nothing.
"Maarten Stekelenburg couldn't hold the ball and the second goal Philippe Senderos played it to their feet and that was a hammer blow.
"When we scored the third goal I thought to really go for it and score the winner but it wasn't to be - we smelled blood and they scored the winner."
Fulham's woes increased when Rodallega limped off with a groin problem which is expected to keep him out for at least a fortnight.
Leicester boss Nigel Pearson insisted he was not getting carried away by the result.
"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."
Source: PA
Source: PA