The England international nodded Iliman Ndiaye's inswinging corner past Nick Pope in the Magpies' goal to open the scoring at St. James' Park after a cagey start to the contest.
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.
Magpies head coach Eddie Howe made six changes to his side for the visit of the Toffees, with Pope, Lewis Hall, Malick Thiaw, Jacob Ramsey, Anthony 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