The Blues' youngsters came out on top over two legs against Aston Villa to clinch a 3-2 aggregate victory at Stamford Bridge.
Villa had taken a 2-1 overall lead in the tie after netting the first goal in west London through Kofi Poyser, but efforts from Marko Mitrovic and Conor Clifford sealed a comeback win in front of more than 12,000 spectators.
Drummy was full of praise for both sides and said the win was just rewards for all the hard work put in at Chelsea.
"You have to give credit to the young lads for playing that way in front of all those people," he told the club's official website.
"To take the ball down and play when you might take a risk and the crowd see you make a mistake takes courage and a strong philosophy.
"They did it admirably. The crowd was absolutely fantastic, they got behind the boys.
"It was a great Youth Cup Final over two legs. Full credit to Aston Villa, they performed very well and scored with a great header.
"The game was played in the right spirit, there was no moaning at the referee and the Villa coaches were very complimentary about us at the end and that was nice.
"I am very pleased for everyone in the academy, especially for (academy manager) Neil Bath who has been here 14 years.
"It is a fantastic Academy and we have got to keep that going."
The victory for Chelsea was the first time they have triumphed in the FA Youth Cup since 1961.