The game was overshadowed by pre-game protests from supporters outside the Emirates Stadium demanding the departure of majority owner Stan Kroenke.
Arsenal manager Mikel Arteta made six changes to the side that drew with Fulham last Sunday, with Bernd Leno, Thomas Partey, Callum Chambers and Eddie Nketiah all handed starts.
A slow start to the game saw both sides struggle to create chances, although Richarlison and Bukato Saka forced the opposition goalkeepers into saves near the half-hour mark.
Gylfi Sigurdsson hit the bar with a pristine free-kick from 25 yards, and that was as good as it got in terms of goalmouth action in a tepid first half.
The second half started with Arsenal granted an excellent opportunity to score when referee John Moss pointed to the spot after Richarlison felled Dani Ceballos in the box. However, the VAR rescinded the decision due to marginal offside in the buildup.
The second half petered into a deserved goalless draw, but Bernd Leno conspired to grant the visitors all three points.
When Richarlison picked up the ball and drove into the area, there was seemingly no danger, as the Brazil international was cornered into an acute angle with no form of support in the box.
His hopeful cutback into the box seemed destined to be gathered by Leno, but the Arsenal goalkeeper somehow mistimed the direction of the ball and hit it into the back of his own net.
VAR checked for offside, but there was to be no let-off for Leno and the goal was allowed to stand.
Gabriel Martinelli stepped off the bench and fashioned Arsenal's best chance of the night in the third minute of injury time, but Jordan Pickford pulled off an excellent stop to deny him.
Up next for the Gunners in the league will be a trip to St James' Park to take on Newcastle United, while Everton will square off against Aston Villa at Goodison Park.
Here are five talking points from the fixture.
#5 Everton pick up first away win over Arsenal in 25 years
Everton have spent much of the last three decades in the shadows of their more illustrious neighbors Liverpool.
It is not for want of trying, though, as the Toffees have regularly invested heavily in their squad, only to be let down by recurring inconcistencies.
The general consensus is that Everton are the biggest 'small club' in the Premier League, too big to be in danger of relegation but also too small to genuinely contest in the upper echelons.
Nowhere is this more evident than in their poor records against the so-called 'top six.'
Whether Arsenal are still deserving of that moniker is subject to debate, but Everton have found the going tough against the north London outfit, particularly away from home.
Prior to Friday, the last time Everton won away at Arsenal in the Premier League was on 20 January 1996 when goals from Andre Stewart and Andrei Kanchelskis helped the Merseyside Blues to a 2-1 comeback victory.
Bernd Leno's own goal meant Carlo Ancelotti inspired Everton to a first away Premier League win over Arsenal in 25 years. It also saw them secure their first double against the Gunners since the 1984-85 season.
#4 VAR makes yet another farcical decision
While the intent and purpose of VAR might be noble, it must be stated that human errors in misinterpretation of situations are making a mockery of the technology's introduction.
There have been several instances of the VAR making shocking decisions, and the game between Arsenal and Everton became the latest to join the list.
For starters, it must be pointed out that the rule concerning the use of VAR clearly states that it shall only be deployed in the case of a 'clear and obvious error' by the referee.
That very phrase goes against everything we see when VARs use micro lines to determine offside decisions. There is nothing 'clear and obvious' about having to determine offside using digitally enhanced lines and images.
The contact between Richarlison and Ceballos that led to the penalty was minimal at most. However, the decision was not chalked off for that reason, but for offside in the buildup.
Nicolas Pepe was adjudged to have gone beyond the last defender in the buildup, but replays showed that his body was seemingly onside, while his arm was slightly beyond.
The Arsenal players were well within their rights to protest the decision, especially considering that the VAR also denied them three points against Fulham.
The introduction of VAR had its fair share of detractors, and this game between Arsenal and Everton gave more reasons why several people are not fans of it.