Villa were sunk by an injury-time winner from substitute Dwight Gayle as Palace edged a tight game.
The defeat was the fourth in a row for Lambert's side and 13th-placed Villa now sit just three points above the drop zone.
The home fans let their feelings be known by booing at the end - while many others had departed before the final whistle - and Lambert understood the frustration among the Villa Park faithful.
He told Sky Sports: "(It was) difficult to take because we had a lot of the ball but we just never created as much as what we hoped for and we got hit with the classic counter-attack.
"When that happens right at the death, it very rarely gives you time to turn it around with a minute to go."
Asked how Villa can dig themselves out of the current run, he added: "You've got to - there's no other way to approach it other than looking up and grabbing it and taking it on.
"If you're going to wilt then it will take you under, you have to meet it head on and that's what we'll try and do.
"You can't criticise the supporters for what they've given ourselves since we came to the club.
"They've stuck with it - it's a huge club with a massive expectance level. We are miles short from where I want to be, that's for sure. It's a huge club but I understand people's frustrations."