The champions had been in danger of dropping vital points to the Barclays Premier League's bottom side before Barry headed home in injury time to snatch a 1-0 win at the Etihad Stadium.
Midfielder 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."
He added: "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."
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.
Royals boss Brian McDermott claimed the decision by referee Mike Dean to award the goal, as well as earlier denying his side a penalty, had ruined his Christmas.
Barry felt he had merely been playing for the ball.
He said: "At the time I felt I just got up and Nicky Shorey was just standing there. I probably went over the top of him. I'm delighted the referee hasn't seen it as a foul."