Jason Demetriou forced in the equaliser to make amends for his penalty miss just a few second earlier.
Another penalty had seen Oldham go in front as veteran Andy Liddell converted his eighth goal of the season, but the high-flying Latics were short of attacking ideas.
Orient hit back strongly and home keeper Greg Fleming had to make an impressive league debut at Boundary Park.
Oldham created the first chance when Danny Whitaker's astute pass found Darren Byfield, who was halted by a last-ditch tackle just as he was about to shoot.
John Melligan fired wide for Orient but Oldham went close when Liddell's drive from a set-piece was blocked two yards out.
Byfield's overhead kick bounced off target but there was an escape for Oldham after their former loanee Sam Parkin flicked on a corner.
Whitaker cleared off the line and the home side breathed a sigh of relief when the loose ball was dragged wide by Adam Chambers.
Orient, however, fell behind on 33 minutes when Tamika Mkandawire grabbed hold of Byfield as the striker attempted to go round him in the box.
Referee Roger East pointed to the spot and Liddell slotted inside Glenn Morris' right-hand post despite the keeper's despairing dive.
Orient responded well with Fleming forced to block Parkin's route to goal and also saved Melligan's raking shot from 22 yards.
After 57 minutes, Fleming showed brilliant reactions to somehow beat out Parkin's close-range volley which looked certain to bring the equaliser.
Parkin was taken off having done everything but score, yet Orient soon received their own penalty as Fleming was judged to have brought down Ryan Jarvis.
After long protests were waved away, the Scot dived to his left and parried Demetriou's low kick, but the ball was fed back in for Demetriou to bundle home from six yards.
Oldham went nearer to a winning goal as Lewis Alessandra and Neal Eardley missed by inches, but the visitors were worth their point.
And Oldham manager John Sheridan admitted: "I have come to the conclusion that some of my players just are not strong enough at home."