The Foxes failed to find the back of the net after a sloppy first half display from the visitors.
Chelsea were much better in the second half following the introduction of Cesar Azpilicueta, Ross Barkley and Mateo Kovacic.
Lampard’s men got their reward when Barkley slotted home a Willian cross on the 63rd minute mark.
The Blues could have added more but Kasper Schmeichel was able to keep them at bay.
In an interview with BBC Sports after the game, the Englishman admonished his men to learn from their mistakes and insisted he was happy with the result but not the performance.
"The result was satisfying and getting to the semis was satisfying,” the 42-year-old said.
“We weren't our usual selves today. We were below what we usually produce. But I am happy with the result.”
“We won the game and must learn from some of the mistakes in our game.”
"I don't want to sound too down after this. You have to show the grit to get a win when you don't play well.”
“We go on to the next one. We can park the Cup and focus on getting the league wins to be in the Champions League."
Chelsea will now face old foes Manchester United in the FA Cup semi-final at Wembley.
