Orient, who won all of their opening eight league games this term, failed to claim maximum points for a second successive fixture, having also been held by Walsall last weekend.
Oldham went ahead through a Charlie MacDonald strike in the 36th minute, with Orient's Shaun Batt firing in the equaliser in the 66th.
Second-placed Peterborough narrowed the gap to the summit with a 2-0 home win over Preston, now fifth, thanks to goals from Tyrone Barnett and Grant McCann.
Wolves, in third, are also only a point off Orient following their 3-0 victory at Colchester.
Leigh Griffiths scored Wanderers' first two goals, the opener from the penalty spot, and Kevin Doyle added another before Carl Ikeme denied the hosts a late consolation by saving a spot-kick from Freddie Sears.
In fourth are Bradford, who were steered to a 2-0 triumph at Walsall by strikes from Kyel Reid and James Hanson.
Rotherham are sixth, having moved into the play-off spot after winning 1-0 at Brentford, courtesy of a Mark Bradley header.
They leapfrogged MK Dons, beaten 3-2 at Gillingham.
Danny Kedwell converted two penalties for the Gills in quick succession early on, with Tom Flanagan being sent off for the foul that led to the second, and a Patrick Bamford brace then had the visitors level by half-time, before Danny Hollands nodded in the winner.
At the other end of the table, Notts County and Coventry, bottom and second-bottom respectively at the start of the day, both moved out of the relegation zone with wins against sides that they sent into it.
County thrashed Crewe - now 21st - 4-0 at home, with Callum McGregor (two), Danny Haynes and Marcus Haber getting on the scoresheet.
Coventry claimed a 1-0 win at Stevenage, who are second-bottom, that was settled by a Leon Clarke goal.
Tranmere are the new basement boys following their 1-0 loss at Swindon, for whom Dany N'Guessan's penalty made the difference.
Elsewhere, Port Vale drew 1-1 at home with Bristol City, with Scott Wagstaff putting the visitors in front and Lee Hughes netting a late leveller, and Carlisle drew 2-2 at Shrewsbury.
The Shrews went 1-0 up through Tom Bradshaw and then came away with a point thanks to Paul Parry's equaliser at the death after goals from David Amoo and Sean O'Hanlan had swung the game Carlisle's way late on.