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."
Fellaini's 93rd-minute header came from Leighton Baines' corner and Lambert admitted he was frustrated by their inability to defend set-pieces this season.
"I am sick of talking about conceding goals from corners," he added.
"They defended so well but Fellaini has scored important goals for them and when the corner comes into the box, Ron (Vlaar) has got him and then he loses him."
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.
"We were ready but we didn't just deal with it that well. I had a strange feeling today [Saturday] we might have to score four goals to win it.
"I hoped I was going to be wrong but it was the way it panned out but it wasn't for the lack of effort from the players.
"With the week we've had - Bolton away in the cup, at home to West Brom and then Aston Villa - if at the start of the week you'd given me a win in the cup and four points I might have taken it."
Fellaini took his Premier League tally to 10 goals for the season while Anichebe scored his fifth in eight league starts having been preferred to Nikica Jelavic for the second successive match.
"We've not really been coming up with the goals but I thought Victor played really well, took his goal well and set up Fellaini's second," added Moyes.
"The two boys both came up with the goods. We are happy with the way Felli got his goals - but I am sure they will be happy how Benteke did for them."