Toffees defender Jarrad Branthwaite had nodded Iliman Ndiaye's inswinging corner in off the upright to open the scoring on 19 minutes after a cagey start to the contest.
Jacob Ramsey's deflected drive then levelled matters on Tyneside after a slow start from the hosts.
But Beto pounced just after the half hour to prod home from close range after goalkeeper Nick Pope could only parry Dwight McNeil's effort from the edge of the box.
Ndiaye lifted a 25-yard effort well over Pope's crossbar early on, while Joelinton curled a shot over after Anthony Gordon's lay-off before Branthwaite struck.
United struggled to carve out any gilt-edged chance in the first half hour, with Joelinton seeing a tame header held by Jordan Pickford and Elanga shooting straight at the England stopper from distance.
After Sandro Tonali picked him out on the edge of the box, Ramsey's drive restored parity - but only momentarily, as Beto's close-range finish put the visitors back in front.
Magpies head coach Eddie Howe made six changes to his side for the visit of David Moyes' men, with Pope, Lewis Hall, Malick Thiaw, Ramsey, Elanga and former Everton forward Gordon all restored to the line-up.
**Newcastle United:** Nick Pope, Kieran Trippier (c), Lewis Hall, Joelinton, Sandro Tonali, Anthony Gordon, Malick Thiaw, Anthony Elanga, Nick Woltemade, Dan Burn, Jacob Ramsey
**Substitutes:** Aaron Ramsdale, Sven Botman, Yoane Wissa, Harvey Barnes, Will Osula, Jacob Murphy, Joe Willock, Alex Murphy, Leo Shahar