James Bailey's stoppage-time volley secured a Coventry a third successive victory in League One as they beat Oldham and moved up to seventh.
Substitute Stephen Elliott put City ahead just after the hour mark, heading home Gary McSheffrey's cross.
Elliott thought he had scored a similar chance 15 minutes later but it was adjudged not to have crossed the line.
Matt Smith headed in an 89th-minute equaliser before Bailey fired in his late winner.
VIEW FROM THE DRESSING ROOM
Coventry boss Mark Robins told BBC Coventry & Warwickshire:
"The overriding emotion was one of relief at the end, because with a minute on the clock left we conceded a shocking goal that would have felt like a defeat.
"At the end of it I'm relieved, and I think we deserved it overall. We were not great in front of goal, and after the efforts of Wednesday you could see they had taken their toll.
"We have to thank the stewards and everybody for clearing the snow and allowing us the opportunity to play. That has given us the chance to win the three points, and it would have been a travesty if we hadn't done."