Arsenal made by far the brighter start against opponents whose lack of confidence was soon apparent, and the visitors were almost ahead after only four minutes.
The chance came when Alexis Sanchez found Mesut Ozil, who freed Aaron Ramsey for a strike that was well stopped by home keeper Jordan Pickford.
Seconds later, Pickford dashed out to close down Sanchez as the Chile international found space near the byline, and Arsenal fashioned a third opportunity after only 11 minutes when Hector Bellerin crossed for Alexandre Lacazette to force Pickford into further action.
But the visitors' early dominance was to prove in vain when, a minute later, Everton took the lead through a superb goal from Wayne Rooney.
Granit Xhaka was closed down quickly inside his own half by Idrissa Gueye, and Rooney picked up possession to curl a great finish past Petr Cech from the edge of the ara.
The goal rattled Arsenal, who were almost two down when Petr Cech was challenged by Dominic Calvert-Lewin but saw the ball run to safety, but the Gunners recovered their composure and levelled the score with five minutes remaining until the break.
Pickford stopped a low shot from Xhaka but was powerless to prevent Nacho Monreal from smashing home the rebound.
Arsenal began the second half on the front foot and took the lead in style after 53 minutes as Sanchez swung in a fine cross from the left and Ozil, arriving in the penalty area, glanced a deft header past the stranded Pickford.
And Everton's hopes of finding a way back into a game in which they had led were soon hit when Gueye lunged in late on Xhaka to pick up a second yellow card as under-fire Koeman looked on in frustration from the touchline.
With 74 minutes played, Koeman and Everton were condemned to another defeat as Arsenal sliced through them again, Ozil finding space down the right and squaring for Alexandre Lacazette, who crashed home an emphatic finish.
Arsenal boss Arsene Wenger brought on Jack Wilshere for his first Premier League appearance for the club since last August, with the midfielder replacing Lacazette for the last 12 minutes, and his side almost added a fourth when Xhaka's powerful strike beat Pickford but not the crossbar.
They did find one after 90 minutes when Wilshere slid a pass through to Ramsey, who slid a neat finish past the demoralised Pickford.
Everton pulled one back when Oumar Niasse struck after a mix-up between Cech and Monreal in stoppage time, but there was still time for Arsenal to add a fifth through a fine angled Sanchez strike.