Everton wasted a host of chances in the first half that ultimately proved decisive with Arsenal goalkeeper Petr Cech making a series of fine saves.
“We are very happy with the goalkeepers. Petr has big experience and big quality. He has started the season very well. He is performing like we want and today he showed us he is very well,” Arsenal coach Unai Emery said.
Arsenal are sixth in the Premier League standings, six points behind leaders Liverpool.
In La Liga, Barcelona salvaged a 2-2 draw against Girona in La Liga at the Camp Nou.
Girona's Cristhian Stuani scored twice while Lionel Messi and Gerard Pique scored once each in a match marred by a controversial red card shown to Barca defender Clement Lenglet.
Barcelona have equal number of points as Real Madrid in la Liga standings.
In Ligue 1, Paris St Germain came from a goal down to beat Rennes 3-1 in Ligue 1. Angel Di Maria, Thomas Meunier and Eric Maxim Choupo-Moting scored a goal each for the Paris club.