Huddersfield were quick out of the blocks and almost took the lead on four minutes when a long goal kick from Simon Eastwood caused havoc in the home defence as O's defender Jordan Spence headed past his own keeper Jamie Jones but was relieved to see the ball drift inches wide.
Booth then knocked the ball into the net when he challenged keeper Jones for a long hanging cross but referee Andy Woolmer penalised the striker for a push.
It came as no surprise when Huddersfield took the lead on 24 minutes when Jimmy Smith's tame header fell into the path of Booth who beat keeper Jones to the ball and then rolled it into an empty net.
Orient's best chance of the first half came on 28 minutes when Jason Demetriou volleyed Morgan's left-wing cross inches wide of the target.
After the break Orient improved and almost levelled when Scott McGleish raced on to Demetriou's throughball - but Town defender Andy Butler got back to block his shot.
Sean Thornton and Wayne Gray both shot wide before Orient levelled on 65 minutes.
Charlie Daniels placed a fine throughball into the path of Smith and he crossed from the left for Morgan to sweep home.
The equaliser was a little harsh on the visitors who had enjoyed the majority of the possession.
In the closing stages both sides had chances to win the game.
Gray fired over for Orient and Daniels tested Eastwood from distance.
McGleish also went close but it was Huddersfield substitute Lionel Ainsworth who almost grabbed a stoppage-time winner only to be denied by keeper Jones.