Ezri Konsa and Gabriel Magalhaes found the back of the net as the Gunners made it seven wins from eight league fixtures to sit alongside City on 21 points.
The defending champions took the lead as early as the seventh minute when Konsa bundled in his goal for Arsenal, after an almighty scramble in the box following a corner.
Arsenal had goalkeeper David Raya to thank for preserving their slender lead, the Spanish shot-stopper keeping out efforts from James Garner and Tyrique George.
The hosts certainly dominated the first half and were unlucky not be extend their advantage, as the visitors still had hope of coming back into the contest.
That flame was dosed after the interval though Gabriel joined his centre-back partner on the scoresheet when he headed in Ben White's cross with little over 20 minutes remaining.
Everton almost pulled one back late in the game when Kiernan Dewsbury-Hall flicked a header just wide.
Arsenal held on for a deserved victory that reaffirms their status as the favourites to retain their league title.
Nicolas Jackson scored twice as Aston Villa beat Manchester City 2-1 to end the visitors' perfect start to the Premier League season. The victory also extended Villa's unbeaten run in all competitions to seven matches.
Nigerian international Alex Iwobi's versatility and quality have earned high praise from Coventry City manager Frank Lampard, who believes the Fulham midfielder was already a good player before their time together at Everton.
Bradley Barcola is available for Liverpool's Premier League meeting with Brighton at Anfield on Sunday, but Dominik Szoboszlai has been ruled out as the Reds continue a demanding run of fixtures.
Arsenal manager Mikel Arteta could hand Max Dowman an increased role in the first team after Bukayo Saka was ruled out for up to eight weeks with a quad injury.
