Villa went into the match on the back of a very different derby experience after beating West Brom 4-3 on Wednesday, and manager Lambert felt they deserved something at Goodison.
He said: "The first goal was really poor from our point of view. I think the second goal you hold your hands up and say it's a world-class free-kick, and you can't do anything about that.
"I don't think we deserved to lose the game. We took the sting out of the crowd and frustrated Everton, who are a good side.
"In the first half, other than Aiden's shot against the post, there's not a save that Guzan's had to make, and we scored a really good goal."