James Tarkowski scored the winning goal for Everton with a header in the second half.
The win is Sean Dyche's first as Everton's new manager.
Everton moved out of the bottom three with the victory.
Bukayo Saka's volley from a corner was cleared off the line by Conor Coady in the first half.
Jordan Pickford also denied Leandro Trossard as Arsenal pushed for an equalizer.
Everton were deserving of their win and improved their performance from previous games.
The result moved Everton up to 17th, a point clear of Wolves and Bournemouth.
Sean Dyche spoke about the need to ignore "outside noise" surrounding Everton.
He encouraged players to provide anonymous feedback on their time at the club.
Everton's start under Dyche bodes well for a more stable second half of the season.