Moncur, on loan from West Ham United, lashed in from a tight angle to ensure the U's claimed a point in their League One curtain-raiser.
Colchester, playing their first home game on the opening day since 2002, made a dream start to the new season when they took a seventh-minute lead.
Magnus Okuonghae, who was appointed as the U's new captain last week, steered a low shot past keeper Paul Rachubka, after Jabo Ibehre’s effort had been blocked.
But Oldham equalised 11 minutes later through Jonathan Forte, who pounced on uncertainty in the U's defence slot into the net.
Ibehre went close for Colchester when he lifted a first-time shot wide with only Rachubka to beat, before Oldham took the lead five minutes before half-time when Forte finished tidily from a pass from former U's skipper Brian Wilson, making an immediate return to Colchester following his summer move to the Latics.
Freddie Sears' shot sailed inches over at the start of the second half, while Rachubka tipped over Gilbey’s superb overhead kick seconds later as Colchester pressed.
Colchester finally equalised just before the hour through substitute Moncur, who lashed in from a difficult angle after Sears’ effort had come back off the post.
The U's pressed hard after that and felt they might have had a penalty when Sean Clohessy's cross appeared to be handled by James Wilson in the area.
Both sides went for a winner after that but an absorbing match finished all square.
For a full match report, pictures and reaction to Colchester's thrilling game with Oldham, see Monday's Daily Gazette.