The Young Foxes set up a clash with either Everton or Southampton after a slender 1-0 win over Chesterfield at King Power Stadium thanks to a 12th-minute strike from Keenan King.
The hosts hit the woodwork three times in the opening half, but it was a solid defensive performance in the closing stages that denied the Spireites an equaliser.
The FA Youth Cup
Fourth Round Proper
Monday 19 January 2015
And a "delighted" Peake was thrilled with his side’s work rate to earn the victory.
"It was a bit more of a struggle than it should have been by not getting a second goal, especially in the first half," he told Foxes Player.
"Fair play to Chesterfield because they came out and had a go. We warned the players that Chesterfield would come out [in the ] second half all guns blazing, so we had to defend quite stoutly at times.
"If one of the shots [that hit the woodwork] went in then the task would have been less difficult. The opposition were buoyed by that and it made them more confident to come at us."
He continued: "In the first 15 minutes of the second half they let us know it wouldn’t be an easy ride. I thought our two centre-halves Cedric Kipre and Elliott Moore defended really well in the centre of that defence.
"It called upon the midfield players to do some tracking back and trying to get back on to the ball as well. There was a lot of work put in to secure that clean sheet."
Elsewhere, Stoke City sealed their spot in the next round with a convincing 3-0 win over Mansfield Town at the One Call Stadium.
Julien N’Goy struck on the stroke of half time and second half efforts from Johnville Renee-Pringle and Daniel Jarvis ensured their victory over the Stags.
It is the first time in three years that the Potters have reached the last 16 of the competition and they now face a trip to Manchester City.
