Moses whose goal drought extended to nine games for the Hammers had a chance to win it for the hosts two minutes from the end, but he volleyed Lanzini’s chip across goal and wide.
The Hammers led through Mauro Zarate’s superb first-half free-kick and had plenty of chances to extend their lead.
But Cheikhou Kouyate, Victor Moses and Diafra Sakho missed decent opportunities while Boaz Myhill made two fine saves to deny Manuel Lanzini.
And they were made to pay five minutes after the interval when half-time substitute Lambert’s shot was deflected in by Winston Reid to earn the visitors a point.
“It’s a fair result over 90 minutes, they deserved something from the second half – but in the first half we should have been more than 1-0 up,” said Upton Park boss Bilic.
“After the goal until the end of the first half we created three or four chances, and their ‘keeper made some great saves.
“At half-time we were pleased with the way we had played. But by four o’clock I was not pleased with the result.”
Bilic’s problems in front of goal were compounded when his main striker Sakho limped off with a thigh injury which could sideline him for some time.
“We are still waiting until the specialists see him, but it doesn’t look good,” added Bilic.
“It’s a big blow for us, he’s a key player.”