City struggled to make the breakthrough in the first half, and the home side were forced to wait until the 61st minute when Carlos Tévez netted the winner.
Gael Clichy cut in from the left wing to tee up the Argentina striker who unleashed a venomous shot past Swansea goalkeeper Michel Vorm – the Dutch keeper was injured trying to stop the shot.
The Citizens right-back Micah Richards pulled up in the second half, with the injuries to the England international and Vorm resulting in a record breaking 12 minutes of added time.
“It’s a massive three points,” said Barry.
“Three games in seven days is always tough. It wasn’t a great performance from us but the manager said at half-time to stay calm and we’ll get a goal.
“And Carlos [Tévez] has come up with the goods – it was a fantastic strike.”
He added: “We have to give Swansea City credit. Their game plan was good. They sat quite deep and made it tough for us to get the ball forward into our players.
“In the second half, we pressed them a bit higher and gradually got on top. But Swansea played a good game but it was a massive three points for us.”
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