Gilmour was handed only his third start in the fifth-round match, having also been selected for a couple of League Cup ties, and seized his chance with an assured performance in a high-tempo tie.
Lampard has given youth a chance this season with Mason Mount, a regular starter, also impressive against Liverpool.
The diminutive Gilmour really caught the eye though with his incisive passing, tackling and fearlessness on the ball.
Lampard could hardly contain the smile when asked about the player snapped up from the Rangers academy.
"What an incredible performance for a young player," Lampard told the BBC. "He was a calm head in that first five or 10 minutes. He's a throwback of a midfielder.
"Can you put your foot in? Yes. Can you make angles to play the passes? Yes. He's only slight in stature but he's huge in personality. He deserves people to talk about him after a performance like that."
They were indebted to recalled goalkeeper Kepa Arrizabalaga to keep them ahead though as the Spaniard, dropped in January, made three saves in quick succession from Sadio Mane, Divock Origi and Liverpool teenager Curtis Jones.
"I was really pleased for Kepa," Lampard said. "It's not easy because all players want to play and a goalkeeper gets scrutinised more than any other position and he made some good saves. He's shown good character and he played well."
Liverpool's third defeat in four games, following Saturday's shock 3-0 loss at Watford that ended their 44-game unbeaten run in the Premier League, can hardly be described as a mini-crisis.
Juergen Klopp's side are 22 points clear in the Premier League and within touching distance of a first league title in 30 years. They will also be confident of overturning a 1-0 deficit against Atletico Madrid in the Champions League.
Yet there are signs that Liverpool are misfiring with eight goals conceded in four games and three consecutive away defeats in all competitions for the first time since November 2014 under former manager Brendan Rodgers.
"Look, it went our way for so long because we defended outstandingly," Klopp told reporters. "Usually you don't get a lot of chances against us and stuff like this, but now we have to admit that in the last three games -- (maybe) four -- we have conceded absolutely too many goals, that's true.
"I am not worried about the momentum -- momentum is not something you get as a present, you have to get it to keep it. We have always a chance to get it back."
He will hope it returns on Saturday against Bournemouth.