The visitors were 3-1 ahead - Christian Benteke hit two either side of a Gabriel Agbonlahor header - but were initially pegged back by Victor Anichebe in the first half. Marouane Fellaini then scored twice, his second coming in the third additional minute.
"I thought we deserved to win the game," said Lambert, whose side have taken just three points from the past eight games since beating Liverpool 3-1 on December 15.
"You are never going to get an easy game here but it is disappointing to only get a point. It is a really tough place to come and three points would have been great but I thought we played ever so well.
"I can't fault the way the lads played and there are not many teams that come up here and beat them. David (Moyes) has a really good side but if we continue to play like that hopefully we will start to get results."
Everton boss Moyes did not think they deserved to lose but was less-than-impressed by his side's defending.
Centre-back John Heitinga had a particularly poor afternoon and he was replaced in a double substitution along with Kevin Mirallas. There were a smattering of boos from the Goodison Park crowd when the Belgium forward was taken off but the jeers which rang out when Heitinga left the pitch were directed at the player rather than the manager.
"I think they were not happy with my decisions," said Moyes of the crowd reaction.
"I think they were the same at Bolton (in the FA Cup last weekend) as well, disappointed with the decisions. I'd rather they direct it at myself if that is the case but we will keep going and trying to earn those victories."
Of the weaknesses at the back the Scot added: "I'd like to say I didn't think we should be in that position (3-1 down) but we were because we defended poorly - the defending let us down. If anyone doesn't think Benteke, (Charles) N'Zogbia and Agbonlahor can't cause you problems then they're wrong and you have to be ready for that."
- Sports & Recreation
- Gabriel Agbonlahor
- Christian Benteke
- Marouane Fellaini
- John Heitinga