The Villans were unable to overturn a first-leg deficit, but the Scot has insisted he will carry on at Villa Park, with the club also currently in danger in the league
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.
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.
"I'm certainly not going to lie down, I've never done it, I'm going to come out fighting. I'm certainly not going to take it on the chin.
"I'm every bit as hurt as the fans are. I know exactly how they feel. I'm absolutely gutted. You couldn't repeat what was said in the dressing-room. Everyone was hurt.
"We created a lot of chances, we played really well in the first half. The atmosphere was brilliant, the crowd were right with us and we've let them down."
Bantams boss Phil Parkinson was delighted with his players' efforts, and believes the side will go down in the club's history forever.
He said: "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.
"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 savour 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.
"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.
"I really feel that this could galvanise 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."