The O's were rocked by a stunning goal by Aaron Cresswell after seven minutes when the left-back beat goalkeeper Jamie Jones with a 35-yard shot, and the nervous home side conceded a second goal on the stroke of half-time.
Midfielder Matt Spring lost the ball in a challenge with Robbie Weir, who played the ball into the path of Adam McGurk and he steered the ball past Jones from the edge of the box.
Any hopes of a second-half revival were left in tatters when Tranmere extended their advantage four minutes after the break.
Cresswell drove a free-kick across the goal and McGurk chested the ball home from close range.
Orient failed to find any inspiration as the visitors enjoyed a comfortable afternoon and they might have added another goal when substitute Zoumana Bakayogo crashed a shot against the post from 20 yards.