Villa's 3-0 home loss means they have conceded 15 goals without reply in their last three games.
Only Stoke striker Cameron Jerome's late equaliser against Southampton prevented Villa from dropping into the bottom three for the start of the new year.
Fans voiced their disapproval with choruses of boos at the end of a calendar year which has seen Villa win only three home league matches.
Lambert, who took over in the summer, said: "The fans since I have been at this club have been brilliant, totally fantastic with myself and the team.
"I can understand them (booing). I would never criticise the crowd here that's for sure, they have been fantastic.
"All we can do is do our very best for this football club. The fans will go away hurt like everybody else, so will we.
"I know that for a fact. But we have to stick together. That's important."
Lambert revealed he intends to hold talks with club owner Randy Lerner ahead of the forthcoming transfer window.
With several players sidelined through injury, it has become increasingly apparent that Villa need to strengthen to bolster their up-and-coming young side.
The euphoria of securing a Capital One Cup semi-final spot and triumphing 3-1 over Liverpool at Anfield just a fortnight ago now seems a million miles away.
When asked about the transfer window, Lambert said: "I will speak with the chairman and see what happens.
"They (the discussions) will remain private, what I say."
But Lambert's faith in Villa to recover from their current poor run of results remains strong.
He said: "The last three results have hurt us, no doubt about it because everyone was positive after the Liverpool game.
"But once three results go against you, there's a negativity.
"I understand that. I can see that. But I have the belief that Aston Villa will be fine.
"You have to keep their spirits up. You have to keep them believing and keep going. That's what you have to do."