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 went in off 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 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 the Croatian. “It’s a big blow for us, he’s a key player.”
West Ham went ahead in the 17th minute when Zarate curled a free-kick up and over Albion’s defensive wall and into the top corner.
But Kouyate glanced Aaron Cresswell’s cross wide, Moses blazed over, Jonny Evans blocked Sakho’s goalbound effort and Myhill twice denied the Lanzini to keep the deficit at one.
And West Ham were made to pay for those missed chances five minutes after the interval when a half-cleared cross fell to Lambert, who fired the rebound straight at the arm of Hammers skipper Reid.
Lambert immediately appealed for a penalty but quickly changed his tune as he saw the ball bouncing past wrong-footed keeper Adrian and into the net.
Moses had a chance to win it for the hosts two minutes from the end, but he volleyed Lanzini’s chip across goal and wide.
West Brom manager Tony Pulis saluted Lambert, who this time last year was still a regular in the England squad but now finds himself kicking his heels on the Baggies’ bench. “We were fortunate to get in the dressing room at half-time only 1-0 down, and then it was time for Rickie,” said Pulis. He’s a great professional. He has not played as much as he would have liked but his attitude is fantastic. The lads want to play and that’ssomething you have to deal with.
“Rickie is a throwback to the times when footballers cared about football, and he cares. He has been a terrific signing for this football club.”
Adrián, Jenkinson, Reid, Ogbonna, Cresswell, Obiang (Carroll 64), Moses, Kouyaté, Lanzini, Zárate (Antonio 82), Sakho (Jelavic 74)
Song, Tomkins, Spiegel, Cullen
Myhill, Dawson, McAuley, Olsson, Evans, Sessegnon (Lambert 45), Fletcher, Yacob, McClean, Morrison, Rondón
Chester, Gardner, Anichebe, Berahino, McManaman, Lindegaard