Germany stand in the way of them booking a quarter-final meeting with either Argentina or Mexico.
Barry, who came through his first 90 minutes against Slovenia since suffering ankle damage in early May, said: "It was a bit of a relief to get through.
"It was a big game and you are always relieved when the outcome is a positive one. We just hope that result can breed confidence with everyone and we can kick on.
"We will be a more confident side on Sunday now. There was a lot of pressure on us. I don't think you would have known the pressure we were under with the way we played.
"We played quite well, created a few good chances and played some good football. Hopefully we can kick on from there. Now we are into the knockout stages anything can happen. We have to believe in ourselves.
"We hope the freedom is there now. We are going into the knockout round and we need to play our natural game.
"You can feel the fear in the other teams. That is probably why there have not been too many stand-out games."