The U's are on a five-match unbeaten run away from home, while Oldham's point saw them move out of the League One drop zone.
Colchester were gifted a 20th-minute lead at Boundary Park when Andy Bond's weak shot was fumbled by home goalkeeper Dean Bouzanis, allowing Drey Wright to fire home from close range.
Both sides created half-chances in a tense contest and Oldham levelled after 68 minutes, with Jose Baxter converting a 10-yard header from Lee Croft's cross for his 14th goal of the season.
Experienced striker Jabo Ibehre went close to restoring Colchester's lead, but his header came back off the crossbar.
Dunne said: "There's still a lot to play for, but I've always been positive that we can get to safety. I'm not happy that we didn't win the game, but this is a decent, battling draw.
"It's all about picking up points from every game, whether that's three points or one, and that's what we are aiming to do.
"I think both sides will be ok at the end of the season because Oldham are looking a different team at the moment and they have two games in hand.
"In the first half we got our tactics right, we got in front and I thought we might go on to score again. We took the game to Oldham, but with the magnitude of the occasion we needed that second goal.
"To be fair, in the second half Oldham took the game to us and were very good. Overall, the important thing is that we didn't get beaten."