City had gone into the game trailing rivals Manchester United by six points at the top of the table and, after failing to make the most of a clear superiority, had looked like handing their neighbours another boost.
Barry said: "I am relieved and delighted at the same time, to get a goal that late. You are starting, probably, to panic, thinking we are going to drop some big points at home again.
"It is difficult, especially when you don't get the breakthrough early. The longer the game goes on the more you could see the Reading players growing in confidence.
"They started to believe they could keep us out for the 90 minutes. Even towards the end they were believing they could maybe get the winning goal."