Villa defeated Norwich 4-1 at Carrow Road in Tuesday night's quarter-final clash to move within one game of a Wembley final.
Lambert's side have also improved in the Premier League after a difficult start to the season and have gone four matches unbeaten in the top-flight.
Martin O'Neill led Villa to the League Cup final in 2010 where they were beaten 2-1 by Manchester United and Lambert is hoping to go one better to give the club's supporters a welcome boost.
"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," said Lambert.
"But the semi-final really gives supporters something to look forward to in the New Year and is brilliant for the club.
"We have had the youngest team in Aston Villa's history, which is an incredible achievement and hopefully shows in the future, given time, that we can build on it and have a good side.
"We are slowly but surely getting there. We're doing really fine. It's another building block."
The victory was tempered by an injury to striker Darren Bent, who suffered a hamstring injury which meant his first start for nearly two months lasted barely half an hour.
"It was a shame because I thought he looked pretty sharp for his first game in a while. Hopefully it's not too bad," added Lambert.