Korey Smith blasted a goal of the season contender in the last minute to secure Latics a vital point in the capital.
Scott Cuthbert had put the O's ahead just before the half time mark but Smith's wonder strike gave Latics a share of the spoils to move the side up one place into 17th.
Both sides started the game equally as they battled for possession in the midfield.
It was Latics that had the first real opportunity of the game on 12 minutes. David Worrall was allowed acres of space on the right hand side to deliver a cross but Jonson Clarke-Harris' header was well saved by the goalkeeper.
The striker was causing a nuisance in attack and ten minutes later was presented with another chance in front of goal. He twisted and turned in attack but his effort was tame and was easily gathered by Jakupovic in the O's goal.
The visitors had a strong grip on the game with captain Smith playing a star role in midfield, comfortably breaking up play and creating attacks.
Orient started to come back into the encounter as Kevin Lisbie and Moses Odubajo tested Latics' defence but were both unable to get their efforts on target.
Latics were hit with a sucker-punch at the end of the half as Orient took the lead against the run of play. Cuthbert got onto the end of a corner by winger Dean Cox and powered a header past Rachubka.
The goalkeeper had to stay alive deep into added time as he prevented the O's from extending their lead, stopping Lisbie with a superb save with his feet.
Athletic started the second half brightly and maybe Clarke-Harris should have done better when allowed space in the area but his header was saved comfortably by the goalkeeper.
On the hour mark, Scott Cuthbert had a golden opportunity to extend Orient's lead but Rachubka stood up well and denied the defender from point blank range.
The visitors didn't let the goal disadvantage get to them and fought for a way back into the game. They went mightily close with just over 20 minutes of the game remaining as a scramble in the area just couldn't be put away by substitute Charlie MacDonald.
Latics were in the ascendancy but weren't really causing the O's goalkeeper any real danger. With 15 minutes of the game to go, Worrall headed a Dayton cross but couldn't get the required power on it to trouble the goalkeeper.
MacDonald, who had been a threat since coming on, was presented with a golden opportunity to equalise late into the game after being put through by Harkins. The striker managed to get in front of his defender but missed the target from eight yards out.
With time almost up, Smith produced a goal of the season contender as he blasted home into the top corner to sent the visiting supporters into delirium.
Latics then defended for the rest of the five minutes added on to seal a crucial point against the third placed side.
Oldham Athletic: Rachubka, Brown, Lockwood, Wilson, Grounds, Worrall (Millar 77'), Smith, Wesolowski (Dayton 62'), Harkins, Clarke-Harris (MacDonald 62'), Philliskirk.
Subs not used: Oxley, Dunfield, Winchester, Gafaiti.
Leyton Orient: Jakupovic, Omozusi, Cuthbert, Baudry, Cox, James, Lisbie (Dagnell 75'), Mooney (Simpson 86'), Odubajo, Clarke, Bartley (Ness 20').
Subs not used: Jones, Sawyer, Lasimant, Gorman.
Goals: Leyton Orient 1-1 Oldham Athletic (Smith 90')
Attendance: 3,645 (238 visiting supporters)