Struggling Oldham appeared to have claimed an unlikely point when Matt Smith met an 89th-minute corner and cancelled out Stephen Elliott's earlier header.
But Bailey smashed home in the dying moments of time added on to ensure the Sky Blues took maximum points.
Robins said: "The overriding emotion is one of relief at the end because with just a minute left on the clock we conceded a shocking goal that would have felt like a defeat.
"But we had that bit of energy to get up the field and that bit of quality to pick out James Bailey and for him to pick out the top corner.
"I thought we deserved to win, but the second-half performance wasn't great, having said that, we have created numerous opportunities that we haven't taken and we weren't great in front of goal.
"We have got to thank the stewards and everybody around the place for clearing the snow and allowing us to play and the opportunity to play.
"That was fantastic because it gave us the opportunity to take the three points and it would have been a travesty had we not done."