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. When you ask managers that, they will tell you they just get on with it. Until you hear anything different you have to run with it.
"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 admitted his side had thrown away a golden chance to reach Wembley.
Lambert's men, trailing 3-1 from the first leg, won 2-1 on the night, but it was not enough and the League Two outfit advanced to the final where they will meet either Swansea or Chelsea.
Lambert said: "I am 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."
Lambert was reluctant to talk about the need to bring in new players before the transfer window closes.
He said: "I will talk about the game. I don't want to talk about January or players coming in. I know the situation of it. I know what's happening."