Villa are also struggling in the Premier League and currently sit 17th in the table but 43-year-old Lambert is determined to turn things around.
"There are two ways to deal with it," he said. "You either accept it or come out fighting, and I will keep fighting because I'm not brought up (to accept it).
"I handle it in my own way, I've got to handle it - you are the custodian of the football club and I will try everything I can to turn it around."
The two cup upsets, particularly that against the Bantams, represent high-profile setbacks but top-flight survival still tops Lambert's agenda, leading him to make several changes last night including resting striker Christian Benteke.
"The lads have played a hell of a lot of games and we've not got the biggest of squads. You have to watch what you're doing," he said.
"You can't make too many changes but the Premier League is the most important thing."
Millwall hit back after Benteke's replacement Darren Bent had opened the scoring for the top-flight side, Danny Shittu heading an equaliser before John Marquis nodded home a late winner at the second attempt.
The result was marred by a number of Millwall supporters causing a halt to the game during the second half as they threw bottles at assistant referee Mark Scholes.
Police made three arrests but Lions boss Kenny Jackett refused to let the scenes overshadow his side's efforts.
"There's a line between supporting your club and creating a passionate atmosphere and we don't want to step over that line," he said.
"We can't condone it and to some degree it is disappointing but on the plus side, the atmosphere was fantastic and it was terrific to be here."
Marquis was making his first start since an FA Cup tie with Dagenham last January after spending most of this season sidelined by a groin injury.
Jackett, who praised his entire team, reserved special mention for the 20-year-old striker.
"I thought it was a terrific performance from the whole team tonight," he said.
"I thought we deserved to win and I was so pleased for John Marquis that he got the winning goal. He has been out for a long time and has worked so hard to get back to fitness."