McNeil's drive gave the Toffees a 79th-minute lead before Doucoure stroked home a second soon after to leave Eddie Howe's men chasing a way back into the contest late in the game.
But substitute Beto's stoppage time finish ended their hopes of an unlikely fightback after what was an uncharacteristic display from the visitors, who were second best at Goodison Park.
The hosts began with purpose, with Ashley Young and James Tarkowski shooting speculatively and waywardly, but United produced the best chance of the opening quarter of an hour or so. Joelinton slid a pass into Miguel Almirón, who found space on the right of Everton's box but shot straight at Jordan Pickford.
Martin Dúbravka, in for his first Premier League start of the season in place of the injured Nick Pope, saved twice from Dominic Calvert-Lewin, whose header was held by the Slovakian before a tame effort was kept out with similar ease. McNeil scuffed wide after a swift break and a United attack led to a chance for Alexander Isak, who glanced Trippier's teasing crossed past the upright.
But the clearest opportunity to break the deadlock in the first half fell to Dyche's men and their number nine Calvert-Lewin. Jarrad Branthwaite hooked a long free kick into his path but somehow, six yards from goal, he blazed over the bar under little pressure from Newcastle's backline.
The Toffees frontman saw his deflected header parried by Dúbravka after the restart and at the other end, former Everton man Anthony Gordon's dipping drive failed to trouble Pickford. Gordon had an even clearer sight of goal soon after having robbed Tarkowski on the edge of his own box, but couldn't find the bottom corner as Pickford grasped the shot.
After some neat work from Isak on the left, Gordon fired over from range, but just over ten minutes from time the Toffees struck. It came following a mistake from Trippier, who was dispossessed by McNeil. The former Burnley winger advanaced before lashing beyond Dúbravka, who was helpless to prevent it.
The Magpies looked to hit back instantly, with Almirón nodding Lewis Miley's miscued shot onto the roof of the net, but it wasn't to be. Four minutes before the end, another error from Trippier allowed Jack Harrison space to drive forward and square the ball for McNeil, who didn't connect with it - but the ball ran kindly for Doucoure, who rolled home his side's second.
And in stoppage time, it got even worse for United. Substitute Beto survived a protracted VAR check after racing through on goal from the right and nutmegging Dúbravka to make it three, rounding off a night to forget for Howe and his charges in the process.
