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."