Lambert's header cancelled out Wayne Gray's opener for the O's in a game that lacked FA Cup drama and magic.
In a dismal first half it was Orient who had the better of the earlier exchanges with John Melligan firing over from the edge of the box in the opening minutes.
On 16 minutes, the O's cashed in on their early dominance. Neat work from Jason Demetriou on the left saw him beat his man and whip in a cross for Gray to head home from eight yards.
Orient continued to dominate possession without creating any chances and Rovers offered little going forward. A stoppage-time header from Sean Rigg that landed on the roof of the net was as close as they came in the first period.
But it was a rejuvenated Rovers who came out after the break, Lambert going close first on 49 minutes when a 25-yard free-kick hit the bar.
On 55 minutes Lambert was played through by the impressive Sammy Igoe, but again sliced his effort wide.
Rovers continued to pour men forward and the game began to open up meaning the O's could counter and they should have stretched their lead on 76 minutes, when the unmarked Adam Boyd headed wide.
Five minutes later they were made to pay for their missed chances. Rovers looked dangerous from set-pieces throughout, and with their eighth corner of the match drew level, Lambert stooping to head home Stuart Campbell's delivery.
Substitute Sean Thornton went close in stoppage time for the O's with a free-kick, but it ended all-square.