The O's needed a controversial equaliser at Boundary Park, but they stayed top of Sky Bet League One with a record of eight victories and two draws.
After the Latics took a deserved lead through Charlie MacDonald, Orient's leveller was awarded despite substitute Shaun Batt clearly being in an offside position.
The match officials ruled that Batt had not got a touch on Kevin Lisbie's low shot, leaving Oldham furious that they were denied their first win in six games.
Orient boss Slade said: "I haven't seen our goal again, but it looked like Kevin Lisbie's shot went straight in.
"The important thing is that we came away with a result. Every week is a challenge and we have to make sure our standards don't drop.
"There was nothing in the game for the first 25 minutes but Oldham got a couple of opportunities and hit the post and took a lift from that.
"They scored during their best period of the game and they deserved to be in front at that point.
"We needed a response because, by our usual standards, the first half had not been great.
"At half-time we were very honest with each other. We knew we were better than that and we talked frankly about what needed to improve.
"We did improve, we showed some character and, after we scored, we looked the more likely winners. We found another gear and it's really encouraging to come away with a result."