Trailing at halftime, Barry struck a fifth goal since December to earn the point.
He said afterwards: "I saw the cross and I was in a good place and finished the goal.
"I was talking before the game and I was happy to do this and score again.
On his slow adaptation to Premier League, Barry added: "I played in Spain last season, so it is not the same competition. I needed to work every day to be ready to score."
Meanwhile, Everton teammate James Garner also said: "We were going into the game hoping to get three but in the first half, we weren't good enough and we needed to react. That is what we did. We'll have to take the point and move on.
"We need to start putting in the performances at home now. We are very good away and we need to start picking up points here."