The win sees Everton leapfrog Leicester into eighth spot, although they are eight points behind seventh-placed Burnley, while Newcastle remain 10th.
Theo Walcott's strike secured all three points for the Toffees in a game with little else to report, but Schneiderlin put in an impressive midfield display.
Everton went close for a second time 30 minutes later. Walcott was originally unable to bring Yannick Bolasie's deep cross under control but the ball came back off DeAndre Yedlin and Walcott smashed it into the roof of the net.
Newcastle could have scored an equaliser in the 64th minute in one of their rare chances in the game. I think everybody is finally going in the right direction.
'I think eighth in the league compared to fifth from bottom shows we are getting them better.
"Apart from the last 10 minutes we dominated the game and there was only going to be one victor".
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Dubois managed to get on the scoresheet with a late goal in Game 6, but that was his only mark over the final three games. Forget the D.C. sports trifecta; you could've made a pretty penny letting it ride on six straight postseason wins.
"We found them very hard to break down because of the very negative tactic of dropping everyone back so you have to be patient and we got the victor so you couldn't knock our football". We've moved on, I've got no gripes against who did it, it's gone, let's not let it happen again, let's move forward. I don't pass the ball'.
"Islam worked hard", Benitez said after the game.
"To be here, to play against this team, and to expect we have to beat them easily, is not right". They have very good players and they can make the difference. We had the chance to draw, maybe even win but we lost against a good team.
Former Newcastle goalkeeper Given thinks Everton fans should remember the situation they were in before Allardyce took the job and be grateful for the current league position.
Towards the end of the season, it gets harder for TV companies to peddle out the Premier League as some sort of flash, new, must-have commodity you can't live without.
The France global has struggled to find his best form since joining Everton from United in a £20m deal in January 2017 and he was jeered by Toffees fans against Crystal Palace earlier this year.