Two goals in 11 first-half minutes from Jimmy Smith and Andrew Whing gave Orient a comfortable half-time advantage, while Harry Kane struck a double late in the second half to heap the pressure on Rovers.
O's midfielder Dean Cox picked out Smith who sent a powerful header past Luke Daniels in the fifth minute and provided the pass for full-back Whing to notch his first goal for the club from 25 yards in the 16th minute.
The visitors improved after the break and were thrown a lifeline when a cross from Jeff Hughes in the 52nd minute was turned onto his own net by Jason Crowe.
Then Will Hoskins and Rene Howe both went close for Rovers and it was against the run of play when the hosts restored their two-goal lead in the 77th minute through Kane.
The on-loan Tottenham teenager collected a centre from Charlie Daniels to nestle a low-shot from 12 yards into the net before the same combination struck again five minutes later before Kane swooped to apply the finishing touch.
Source: PA
Source: PA