Villa reached the 2010 final of the competition, losing 2-1 in controversial fashion to Manchester United, but since then their fortunes have nosedived, including several managerial changes.
But there have been signs of improvement and a four-game unbeaten league run was followed up with Tuesday's 4-1 win at Norwich to secure a place in the Capital One Cup semi-finals. And Lambert said: "The cup win is a huge win for us and to have a semi-final, with full houses home and away, will be brilliant for the club."
Lambert has put the accent on youth - the average age of the squad is less than 24 - after spending £22million in the summer and Villa are facing a battle to preserve their top-flight status.
The Villa boss added: "I think we are an easy target for people saying we are a young side and all that, but the reality is we've been playing very well despite a tough run of fixtures.
"We've come through a really hard run with the two Manchester clubs and Arsenal in successive matches. That's really, really demanding with a young team. They need a bit of help but if you can give them time, and keep that collective spirit with the crowd we've got, they'll drive us on.
"The club has had a lot of heartache for the past two years or so and fans haven't had a lot to shout about. But the semi-final really gives supporters something to look forward to in the New Year and is brilliant for the club."
Villa are assessing the extent of the hamstring injury suffered by striker Darren Bent which meant his first start for nearly two months lasted barely half an hour at Carrow Road. Lambert said: "It was a shame because I thought he looked pretty sharp for his first game in a while. Hopefully it's not too bad."
But there was again evidence of Lambert having other firepower to call upon via two-goal substitute Andreas Weimann and Belgian international Christian Benteke, who netted the final goal.
Lambert said: "One thing about Andy Weimann is that he is a natural finisher. I left him out just to give him a rest, because he had run himself into the ground in the two Manchester games and against Arsenal, but he is a natural goalscorer. He knows the game pretty decent for someone so young and certainly knows where the goal is, that is for sure."
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