Villa looked like pegging back the League Two side's 3-1 lead from the first leg when Christian Benteke scored in the 24th minute at Villa Park.
The hosts dominated the first half but James Hanson equalised on 55 minutes and Bradford held on in a tense finish after Andreas Weimann's 89th-minute strike.
Lambert, who said he would not walk away following the result, said: "I'm absolutely gutted, disappointed, hurt, everything.
"Everyone is hurt. You will never have a better chance to get to a cup final throughout your whole career, even the young lads.
"It may take them 10 or 12 years to get even close to one again. That's the hurt.
"I am every bit as hurt as what the fans are. It's my responsibility. I know exactly what it's like and I know exactly what they're feeling because I am feeling the exact same.
"We had two chances to do it and never managed to do it. At 1-0 you think you are going to do it and then we lose it.
"It was a really poor goal from a corner. Over the two games that's four goals from set-pieces which is certainly not good enough."
Bradford fans chanted 'you're getting sacked in the morning' but Lambert is adamant he remains the right man for the job of rebuilding Villa.
He said: "Am I still the man for Villa? Yes, absolutely. There are two ways - you either lie down and take it or you come out fighting. I am certainly not going to lie down."
Lambert will now have to lift his team for Friday's trip to Millwall for an FA Cup fourth round tie.
He added: "We've got another tough game at Millwall on Friday but you have to pick yourselves up and go again."