James Hanson's goal was enough to secure a place at Wembley for the Bantams, despite efforts from Christian Benteke and Andreas Weimann for the hosts, who won the second leg of the semifinals 2-1 but fell on a 4-3 aggregate.
The Villans have also struggled in the Premier League this season and currently sit just above the drop zone, having won none of their last six games.
When asked if he would continue as Villa boss, Lambert told reporters: "Yes, absolutely.
"There's two ways you can respond. You either lie down and accept it or come out fighting. We are certainly not going to lie down."
On the other end of the spectrum, Bantams boss Phil Parkinson was delighted with his players' efforts, and he believes the side will go down in the club's history forever.
"These lads will be remembered in the history of Bradford City for years to come, there's a 1911 lounge at the club to celebrate the cup victory of that year," Parkinson said. "In years to come there'll be a lounge named after this cup run and these players because of what they've achieved.
"It's a moment to savor for everybody, the supporters, the players and myself. We've had our families in the crowd tonight, so for them to be a part of what we've achieved tonight is something we'll never forget."
Bradford City will play either reigning UEFA Champions League winner Chelsea or Swansea City in the final at Wembley Stadium
"To go to Wembley is going to keep the club going for quite a while, for the city of Bradford I think it's massive," Parkinson said. "I really feel that this could galvanize the area, our supporters have stuck with the club through some really tough times, and I'm so pleased tonight that we've given them something to go into work tomorrow and hold their heads up and be proud of the club.
"We'll play [the final] down and go as underdogs again and we'll give it a good go and enjoy it."