The visitors went ahead in the 10th minute when skipper Stephen Dawson fired into the top-right corner after Rovers failed to clear the ball properly.
Then in first-half stoppage time Orient doubled their lead when Scott McGleish found the back of the net. Scott Wootton pulled a goal back for Les Parry's side but Orient held on for three precious points.
After Dawson had opened the scoring, disaster struck in first-half added time for Rovers when goalkeeper Gunnar Nielsen spilled a shot by Paul M'Poku, presenting McGleish with an easy finish.
Tranmere were back in contention in the 61st minute as Wootton turned an Aaron Cresswell free-kick past an unsighted Brown. Then Jason Brown produced a tremendous reaction save to deny Dale Jennings an equaliser.
Showunmi was next to be thwarted by the alert keeper, who then saved brilliantly from a misplaced header by his own man Aaron Brown. Tranmere were again out of luck after 84 minutes when an Ian Goodison shot hit the bar.
Source: PA
Source: PA