Don’t be fooled by Wayne Rooney’s late spot-kick because the struggling visitors were lucky to escape with a point here.
As they so often have in recent weeks, they dominated possession without looking remotely like scoring in a month of Sundays, and then went behind.
This time it was Anthony Knockaert doing the damage, and there appeared to be no way back until Brighton skipper Bruno handed them an unexpected last-gasp lifeline.
His elbow did connect with Dominic Calvert-Lewin, but it was a soft call by referee Michael Oliver, and Rooney took full advantage to send Mat Ryan the wrong way from the spot.
Amazingly, Everton could even have won it in the fourth minute of injury time when Ryan stuck out a boot to deny substitute Kevin Mirallas from point-blank range.
But it is 12 away games in the Premier League without victory for Koeman’s side now, and they have scored just seven times in their last 13 top-flight matches.
Koeman is a manager still searching for something, anything, from an expensively-assembled but rag-tag team full of round pegs in square holes, with no identity and little or no confidence.
He remains safe for now, despite Brighton fans singing: “You’re getting sacked in the morning” when their team was in front, but he has Rooney to thank for sparing his blushes in this one.
With the two lone strikers starting for each team boasting a combined age of 65 it was perhaps no surprise that goalmouth action was as a rare as snowflakes in the desert in the first half.
Rooney and Glenn Murray found chances few and far between, and defensive midfielder Idrissa Gueye was the only player from either side to manage a shot on target.
He was fed the ball in space by Everton team-mate Nikola Vlasic and let fly from 25 yards to force a diving save from Ryan. It was the only thing he had to do for 45 minutes.
Brighton had struggled to score goals even before joint top scorer Tomer Hemed copped a ban for stamping on Newcastle’s DeAndre Yedlin.
But they looked especially toothless trying to play on the counter against an Everton side who dominated possession but rarely threatened themselves.
The home side could have had a penalty when Michael Keane appeared to block a shot from Lewis Dunk with his hand, but referee Oliver waved play on.
Things did not improve much at the start of the second half either, although Mason Holgate, on a rare surge forward from right back, did force another save from Ryan.
Calvert-Lewin also had a tame header saved as Everton showed glimmers of promise, but for a £45m signing, Gylfi Sigurdsson offered almost nothing for the visitors.
Gueye was in the wars all game, and Davy Propper was perhaps lucky to get away with only a yellow card after standing on his ankle.
Koeman threw on Oumar Niasse to try and liven things up but it was Calvert-Lewin who had the next chance, chesting down a far-post cross only to shoot tamely at Ryan.
Jose Izquierdo, on for Solly March, finally had Brighton’s first shot on target in the 75th minute but Jordan Pickford turned it away.
Everton then had Michael Keane, outstanding in this game, to thank for blocking Anthony Knockaert’s goalbound volley.
But he could not do anything about Brighton’s 82nd-minute opener. He did block Izquierdo’s initial effort, but Everton failed to clear and Knockaert smashed it home.
The Amex erupted after that, and it could have got even better for the Seagulls when subsititute Izzy Brown forced another fine save from Pickford.
But while the drama was still to come with Bruno handing Everton a lifeline and Rooney scoring a big, big penalty and Mirallas almost snatching a winner, the same questions remain for Koeman.
