Villa's season is approaching crisis point after their shock exit at the hands of the League Two side on Tuesday night.
Villa won 2-1 in the home second leg but went out 4-3 on aggregate as they squandered the chance to salvage a season which has seen them embroiled in a Premier League survival fight.
Now they face another potential upset in Friday's FA Cup tie at Millwall before entertaining fellow relegation candidates Newcastle on Tuesday.
Lambert is believed to be under no imminent threat for his job and only last week sources close to the club were adamant that Villa "needed stability and continuity rather than change."
But the pressure will inevitably increase should they suffer another cup exit to lower league opposition and then lose to the Magpies, who themselves have been in free-fall.
When asked if he was still the man to take Villa forward, Lambert said: "Yes, absolutely. When you ask managers that, they will tell you they just get on with it until they 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. I have never done it. I will come out fighting."
Lambert knows Villa have thrown away a golden opportunity to secure a second Wembley final in the League Cup in four years.
As in the first leg, a set-piece led to their downfall as Bradford striker James Hanson headed the crucial goal which divided the teams.
Lambert said: "Everyone is hurt because 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.
"In the first half, we looked really fine, we looked good, we got a goal and you thought we'd go on and get the chances.
"But over the two semi-final legs we have lost four goals from set-pieces. That's where the hurt is."
Christian Benteke opened the scoring on the night and substitute Andreas Weimann's late goal put Villa back in front after Hanson's intervention, but Bradford held on.
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