The result brought to an end Cheltenham's five game unbeaten revival and lifted the O's up to 15th in the table.
Cheltenham remain rock-bottom and could be relegated by the end of the Easter weekend.
Despite their problems, the home side began the game brightly with their new-look team of young players taking the play to Orient and on-loan Bristol City midfielder Frankie Artus prominent in most things they did.
Striker Elvis Hammond shot wide and winger Michael Antonio went close with a firm volley, but the visitors took the lead against the run of play with 25 minutes played.
The chance came from a long downfield kick from goalkeeper Jamie Jones that was flicked on by Scott McGleish.
Jason Demetriou guided the ball into the path of Dean Morgan and he beat Scott Brown with a low finish into the far corner of the net.
Orient very nearly made it 2-0 four minutes into the second half when Morgan fed McGleish inside the penalty area with only the goalkeeper to beat.
McGleish got his shot past Brown but defender Michael Townsend got back to make a block on the line.
The rebound fell to Simon Church but Townsend somehow kept his shot out as well.
Cheltenham huffed and puffed during a second half of little incident but had the best chance of the remainder of the game when diminutive midfielder David Hutton skilled his way into the box but saw a left-foot shot touched over the bar by Jones.
Cheltenham were also convinced that they should have had a penalty five minutes from the end when Andy Gallinagh's cross appeared to be handled by James Smith inside the area but referee Gavin Ward was having none of it.