Despite the wet and windy conditions, both sides played their part in an entertaining contest and visiting defender Jeremy Gill almost put his side ahead on three minutes but O's midfielder Jason Demetriou smuggled the ball off the line.
Orient gradually took control and Paul Terry had a penalty claim turned down before Charlie Daniels fired in two fierce shots that were deflected away for corners.
Orient's top-scorer Boyd had a header saved by Shane Higgs and the visiting keeper then made a stunning stop to deny the O's marksman before Wayne Gray slid an effort against the post.
Cheltenham had a couple of efforts from the livewire Steven Gillespie but both were saved by Glenn Morris before Orient took the lead on 41 minutes.
Gray whipped in a cross from the right and Boyd headed home his eighth league goal of the season from six yards.
After the break, Gillespie continued to cause the home defence some anxious moments but Orient should have stretched their advantage when Adam Chambers saw his shot blocked and the rebound fell invitingly for Boyd, who swept his attempt over the bar from four yards.
However, Cheltenham were just as wasteful and former Orient striker Paul Connor should have levelled in the 72nd minute and then Morris saved well from Gillespie.
It proved to be a costly miss as four minutes later, Orient wrapped the game up.
Demetriou's deflected cross from the right fell invitingly for Orient substitute Melligan to head home from close range and sink his former Cheltenham team-mates.