Barry netted early in the second half to lift Everton into seventh place in the Premier League and leave Newcastle in 13th.
Newcastle midfielder Jacob Ramsey was sent off in the final seconds for a crude foul on Harrison Armstrong.
It was a fitting end to another painful defeat for Saudi-funded Newcastle, who have lost four of their past five matches in all competitions, winning just twice in their past seven.
Jaissle and his players were booed off at half-time and again after the final whistle as the German coach begins to feel the heat in his first season in charge.
Hired following Eddie Howe’s surprise close-season exit, former Red Bull Salzburg and Al-Ahli boss Jaissle has struggled to make an impact on Tyneside.
Jaissle desperately needs a win at fellow strugglers Fulham on Saturday to ease the mounting scrutiny.
Bizarrely, Newcastle have lost 22 of their past 33 Premier League matches on a Monday and many of their supporters headed for the exits well before the final whistle on a miserable rainy night.
Barry’s eighth goal this season, including four in his past six matches, gave Everton their second win against Newcastle in the space of five days following their 3-1 victory in the League Cup fourth round on Thursday.
Everton ended their run of two successive league defeats after losing against Arsenal and Chelsea.
It was Everton’s first away league win since February 28 when they took three points on their previous visit to Newcastle.
Jack Grealish was absent from the Everton squad for a second successive game after being axed for the Newcastle match last week owing to a “breach of club discipline”.
The on-loan Manchester City winger, who has been criticised for his off-field behaviour for much of his colourful career, reportedly skipped training before the League Cup clash.
Everton said he was “not match fit” for the rematch with the Magpies, but his team coped just fine without him.
Newcastle made a lively start and went close when Harvey Barnes fired just wide before Joelinton was narrowly off target from 25 yards.
But chances were at a premium for both sides in a scrappy first half more notable for perspiration than inspiration.
Everton’s opener four minutes after half-time was in keeping with the grimy fare that preceded it.
Newcastle’s dismal marking let in Tyrique George for a shot that drew a weak save from Lukas Hornicek.
Hornicek could only push it straight into the path of Barry for a simple tap-in that survived a lengthy VAR check for offside.
It was a chance so easy that even the notoriously erratic Barry couldn’t fail to hit the target, although Newcastle were aggrieved by the VAR decision to let it stand.
Fabian Schar wasted a good opportunity to equalise when he blazed wide.
And Jaissle’s men came within a whisker of levelling through Anthony Elanga, whose close-range shot clipped the post and rolled along the goalline before Everton scrambled it clear.
Summing up their inconsistent display, it took 86 minutes for Newcastle to finally register a shot on target and even then Yoane Wissa’s effort was routine for Pickford.
Wissa did manage to force a better save from Pickford in stoppage time.
But Ramsey saw red for two bookings in the space of 60 seconds to complete Newcastle’s misery. (AFP)