The defeat at Goodison Park was Everton's fifth in their last seven Premier League games and left Koeman's position looking increasingly precarious as they dropped into the relegation zone after a humiliating day, having spent a reported £142million in pre-season.
It looked as though it was going to be a long day for Koeman right from the start, as Arsenal picked his side apart early on and the visitors should have taken the lead twice before Wayne Rooney rolled back the years with a long-range strike.
But such was Arsenal's dominance that an equaliser was inevitable and it eventually came from the unlikely source of Nacho Monreal.
Alexis Sanchez then set up Mesut Ozil for their second just after the break, before Idrissa Gueye's dismissal for a second booking left the Toffees with a mountain to climb with 22 minutes to go, and Arsenal ultimately blew the hosts away.
Alexandre Lacazette added another to round off a flowing move, before the excellent Aaron Ramsey put the match beyond the beleaguered hosts with the first of three goals in stoppage time.
Substitute Oumar Niasse did pull one back late on, but Sanchez restored the three-goal cushion with his first of the season to move Arsenal up to fifth on Arsene Wenger's 68th birthday and compound a miserable day for Everton and Koeman.
Wenger named a very attack-minded starting XI and that positivity inspired a vibrant showing early on, with three chances coming inside the first 10 minutes.
Ramsey spurned the first, seeing his effort tipped wide by Jordan Pickford after a fine Ozil pass, before the goalkeeper also denied Lacazette from a tight angle.
The Frenchman should have opened the scoring with his second chance, but Pickford was there again to deny him from close range after the striker pulled off a lovely turn.
Everton capitalised on Arsenal's profligacy completely against the run of play shortly after, however.
Gueye robbed Granit Xhaka just outside the area and Rooney duly struck a sumptuous effort into the top-left corner from 25 yards.
But Arsenal's persistence finally paid off five minutes before the break - Monreal smashing a low shot beyond the goalkeeper after Xhaka's previous effort was parried.
Koeman injected a little more energy into his midfield by bringing Tom Davies on for Ashley Williams at half-time, but Arsenal remained the more purposeful going forward.
They completed the turnaround eight minutes after the break, with Sanchez clipping a delicate ball into the danger zone and Ozil was there to head past Pickford at the end of a swift break.
Things were made even more difficult for Everton when Gueye picked up a second booking for a rash tackle on Xhaka 22 minutes from the end, with the Gunners taking advantage soon after as Lacazette swept home from 12 yards.
Ramsey opened up a three-goal lead in the 90th minute with a cool finish following an incisive pass from substitute Jack Wilshere.
A defensive mix-up between Monreal and Petr Cech allowed Niasse to pull one back in calamitous circumstances a few moments later, but Sanchez ensured Arsenal had the last laugh, finding the bottom-left corner with an excellent finish to leave Everton battered and bruised.
