Gilmour had surgery last Friday after suffering a knee injury during the 3-2 victory over Crystal Palace and he is expected to be out for three months.
The 19-year-old will now have a period of rest before he begins his rehabilitation and Lampard is confident he will be back as quickly as he can be.
Lampard said: “It went as well as it possibly could. He was in the training ground on Sunday on his crutches. He is going to keep his feet up.
"His mum is coming down to help him, probably cook, clean and all those things, his mum and dad will be there to do that.
“I am confident with Billy he will do everything right and be back as soon as possible.”
Chelsea play Norwich on Tuesday and Lampard has called on his players to be more vocal on the pitch after their meek 3-0 defeat at Sheffield United on Saturday.
Lampard said he could only hear Sheffield United voices at Bramall Lane and has spoken to his players about being louder.
He said: “It is something I talk about with the players. I think is one of the easiest things to do on the pitch to help your team-mates and yourself but being vocal, driving people on, giving them information, talking.
“We have to get better at that and that is what I drill into the players. Particularly when you can hear it now, the stadiums feel different and it something we need to be better at.”