Paul Konchesky's drilled effort in first-half stoppage time was enough to seal a vital win for the Premier League's bottom side, while Villa drew a blank for the 12th time this season.
Despite insisting he was happy with the other aspects of his side's game, Lambert was forced to admit a lack of goals is becoming an increasingly pressing concern, with Villa now without a win in their last six league games.
Lambert told Villa's official website: "We never had enough in the final third. We had a lot of the ball but in the final third where it mattered we just didn't produce it.
"We have to put that right. We have to do better in the final third of the pitch."
The former Norwich City boss came in for plenty of criticism from the stands, with many Villa fans voicing their disapproval at another poor performance from their side.
Yet Lambert insists that he and his players are committed to turning their form around and kick-starting their season.
He added: "You have just have to keep working hard, keep working hard at the training ground, keep everyone upbeat and keep the lads believing that it will come.
"As I said to them in the dressing room, the effort and commitment was there but you just need to be better in the final third.
"We need to find a way to get our good play goals."
A miserable afternoon for Villa took a turn for the worse late on when Ciaran Clark was shown a second yellow card for squaring up to Leicester's Matty James, who received a straight red for a late tackle on Jores Okore seconds earlier.
Lambert was at a loss to explain Clark's dismissal.
"I saw the red card [for James] and I thought it was a red card," Lambert added.
"I don't know what Ciaran has been sent off for. Maybe the referee viewed it as a second yellow because Ciaran got involved. I don't know.
"I thought the tackle was pretty high."
