The striker struck in the 49th minute after goalkeeper Lukas Hornicek failed to hold Tyrique George's effort, allowing Barry to finish from close range. Everton had been the more purposeful side for much of the match and deserved their victory, while Newcastle struggled to create clear opportunities.
David Moyes was pleased with his team's performance and felt they could have won by a greater margin.
"We had a couple of really good games against Newcastle recently and got some good results. I think we deserved it overall. We were better and should have got a few more goals," the Everton manager told BBC Match of the Day.
"In the end we were able to deal with it and see it through. Keeping a clean sheet is a big part of what we try and do."
Everton had created the better chances before the break, with Newcastle United unable to establish their usual attacking rhythm. Moyes also highlighted the importance of scoring first after his side had failed to make their early opportunities count.
"Lots of the game it was [one of their best performances]. We probably had better chances in the first half. [Not taking them] kept it really difficult. Getting the first goal was important," he told Sky Sports.
Barry's goal came shortly after the interval, with George's shot slipping beneath Hornicek and the Everton forward reacting quickest to turn the ball over the line.
"We're thrilled he's getting the goals because you need your centre forward to score. He has missed a few but thankfully tonight he was on the spot to get the goal," Moyes said.
Newcastle attempted to respond but found it difficult to trouble Jordan Pickford, who produced an important late save to deny Yoane Wissa and preserve Everton's advantage.
Moyes praised the goalkeeper's contribution, although he suggested such interventions had become an expected part of his performances.
"He's had some brilliant saves for us. The only compliment I can give him is that it's what we expect from him. He's maturing and is a lot calmer in a lot of his actions. He's maturing with age as well."
Newcastle's frustration was clear at full-time, with supporters booing a performance that Jaissle admitted had fallen well short of the required standard.
"Unfortunately we couldn't deliver actually over the whole 90 minutes it was not our game and our style of playing. That is something we are very frustrated and disappointed about," the Newcastle manager said.
"If you cannot deliver when it counts then you don't deserve to win. We were not brave enough in possession, too many long balls."
Jaissle was also unhappy with the build-up to the goal, believing Everton should have been penalised for a foul before the visitors entered the penalty area.
"In my eyes it is clearly a foul so a disallowed goal. Everything comes together. There's a foul then we are not awake and they can penetrate our box. Also a referee decision where I think he was wrong."
The Newcastle boss accepted responsibility for the performance and acknowledged the scale of the work required before their next match against Fulham.
"After a performance like that I can understand [the booing]. That's the reality and I am responsible for that. We need to work on the things we need to improve and after today that is a lot."
Newcastle's night ended with further frustration as Jacob Ramsey was sent off in stoppage time after receiving two yellow cards in quick succession. Everton held on to secure another victory over the Magpies, while Moyes said winning twice against the same opponents in such a short period was particularly valuable.