The win moves Everton up to fourth in the table, and Barry is hoping that his side can maintain their strong run of form.
"The first half, we started so well, with a fantastic tempo," Barry told the media. "We kept the ball well, we created so many chances and it could have been four or five.
"Fair play to Newcastle, they came out and made a game of it, as we thought they would, but we were slightly disappointed with the second-half performance.
"It's another big three points. We're still unbeaten and we want to keep that going as long as possible."
The match marked Barry's 500th Premier League appearance.