No wins in their last five matches, just two victories in the last 13, eight points from nine games and only six league goals add up to a whole host of problems for the Dutchman.
Even Wayne Rooney's recreation of his famous breakthrough goal as a precocious 16-year-old against the same opposition and in the same Goodison Park net to give Everton an unexpected early lead could not mask the issues the Toffees face.
It was an advantage they never looked like holding as Nacho Monreal equalised just before the break with Mesut Ozil, Alexandre Lacazette – after Everton midfielder Idrissa Gueye was sent off – and Aaron Ramsey hammering more nails into Koeman's coffin.
Substitute Oumar Niasse pulled one back thanks to poor defending but fittingly Alexis Sanchez had the final say with the last kick of the game as Arsenal's eased to a first away win of the season.
How the Dutchman, who spent £150milllion in the summer, must wish he could call on the pace of a player like Sanchez, the creativity of Ozil – both of whom are not even guaranteed to be at Arsenal next season – or the finishing of Lacazette.
The three caused problems from the third minute with the overworked Jordan Pickford tipping Ramsey's shot around the post after Sanchez and Ozil combined to cut Everton wide open through the middle.
It was a recurring theme for the England goalkeeper, comfortably Everton's man of the match, as he went on to deny Sanchez (three times), Lacazette (twice) and Granit Xhaka in the first half.
Ozil, meanwhile, created more chances (six) than all the other 21 players on the pitch.
Unfortunately for him Pickford's save from Xhaka led to Arsenal's 40th-minute equaliser as he could only parry the low, deflected shot into the path of Monreal, who scored the club's 1,500th Premier League goal under Arsene Wenger.
It was the culmination of a build-up of pressure, as after Rooney's goal it was virtually attack versus defence.
There was a nice symmetry about the former England captain's 12th-minute strike, however.
It was his 12th against Arsenal (nine for Manchester United and three for Everton – including that famous first in the Premier League) and meant he has now found the net against the Gunners in 12 separate calendar years.
Gueye, playing with more effervescence having been given the solitary defensive midfielder job, dispossessed Xhaka and Rooney did the rest.
Everton's Senegal international was joined in midfield by Tom Davies for the second half after Koeman switched to a back four with the removal of Ashley Williams intending to shore up space – but it left a gap at the back and Ozil nipped in to head home Sanchez's near-post cross in the 53rd minute.
The Chilean, operating on the left, gave Everton academy graduate Jonjoe Kenny a torrid time on his first Premier League start but the 20-year-old was not afforded much protection from more senior team-mates.
A difficult situation was made worse when Gueye, after a first-half booking on Sanchez, was shown a second yellow card after going in late on Xhaka.
Lacazette made it an impossible task from a cross by Ozil, exploiting space down the right, in the 74th minute, extending Arsenal's lead to 3-1.
Koeman introduced Oumar Niasse for Dominic Calvert-Lewin, drawing boos from the crowd, and Ademola Lookman for Rooney but the game was already beyond them and a late flurry saw Ramsey and Sanchez score either side of Niasse's consolation.Let's