The 19-year-old underwent surgery on Friday after injuring his knee in the 3-2 win over Crystal Palace last week.
Gilmour is expected to miss three months of football.
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."