Although the Gunners enjoyed more possession and created more danger than the Toffees, the indigestion was still there among Arsenal supporters as the visitors came close several times to drawing level. Stout defending saw off the challenge and sent Arsenal, perhaps temporarily, to the top of the Premier League table.
Here are three things we learned from the match.
This Arsenal side can dig deep
Arsenal's third intense match in a week's time required considerable fortitude. The conditions were unpleasant, to say the least. Rain made the ball slick and the pitch slow.
Everton played with the energy and intent of a team with something to prove after the 3-0 home defeat to Manchester United last week. Their physical and mental readiness wasn't affected by midweek European action, either.
The Gunners, meanwhile, were four days removed from a grueling, emotional win over Bayern Munich, which came after a patient but taxing victory at Watford.
Arsenal showed the effects of this schedule--the fullback pair of Hector Bellerin and Nacho Monreal in particular seemed to flag--but in the moments that mattered, the concentration and performance reached a high level.
Central defenders Koscielny and Gabriel were especially effective during critical passages of play, intercepting threatening passes for Everton striker Romelu Lukaku. We'll not soon forget Gabriel's late tackle away from the Belgian and primal scream that showed his desire to finish the job.
The intervention of Cech was also crucial. His 86th-minute scamper to block Gerard Deulofeu's open shot prevented the dangerous but divey Spaniard from yet again snatching a point for his team at the Emirates.
A goalkeeper with that kind of focus must provide both an example and confidence for the rest of the team.
Cazözil is a distinctively spectacular midfield combination
Maybe this match didn't teach us about the skill and influence of Arsenal's two midfield geniuses. Most knowledgeable supporters and observers already recognized the special qualities of Santi Cazorla and Mesut Özil.
Against Everton the duo's mastery was on full display.
Each notched an assist with perfect passes into the dangerous area in front of Tim Howard's goal. Those were two of the ten key passes, five each, that Cazorla and Özil made during the match.
That's strong evidence of their incisiveness. In the attacking third, Cazorla completed 28 of 33 passes (85 percent), while Özil connected on 30 of 34 (88 percent). Unbelievable efficiency in the most crowded part of the pitch.
The pair also worked together to propel Arsenal forward. According to FourFourTwo's StatsZone app, Cazorla and Özil were involved in the match's two top passing combinations, finding each other a total of 36 times.
The offensive flow that Arsenal achieves owes to the influence of these two remarkable players.
Arsenal's match management is a work in progress
It's a test of the physical, mental, and psychological, individually and collectively, to succeed consistently at the high level at which Arsenal competes. Naturally, the players and the team don't earn perfect grades on every such examination.
On Saturday, Arsenal showed it has room for improvement in managing the dynamics of a match. Instead of consolidating its lead by placing priority on defense just before halftime, the Gunners kept moving forward. A perhaps ill-advised shot by Alex Oxlade-Chamberlain helped Everton launch a dangerous counterattack, from which an unlucky deflection brought Ross Barkley's goal.
The thought could have been to score a third goal before the break in order to demoralize Everton and put the game beyond much doubt. But the risk associated with that decision, especially against a team so effective in open spaces as Everton, probably outweighed the reward.
What's encouraging is that the lesson did not prove damaging. And the match showed Arsenal's capacity for another type of victory.
If Arsenal add set-piece goals to the repertoire, they'll be devastating. The Gunners have three goals from corners, crosses and throw-ins in this Premier League season, half as many as the top performers in that category, Leicester City, West Ham, and Crystal Palace. (Stats from Opta via Squawka.com)
We know how effective the Gunners can be on the counterattack, with intricate passing moves, and with individual moments of brilliance. The high expected goals stats are evidence of that. Delivering more set-piece goals would make Arsenal a diversified, and difficult to stop, offensive force.