Koeman watched his side limp to a 2-0 aggregate win over Ruzomberok to set up a play off tie with Hadjuk Split in the Europa League with an alarming lack of firepower.
The Toffees hold a slender 1-0 advantage thanks to Leighton Baines' strike at Goodison Park last week in a fixture that saw Pickford start on the bench.
Giacomo Bonaventura and teenager Patrick Cutrone scored the hosts' goals against the Romanian visitors. It was a physical game and 90 minutes that the players needed.
The only goal was scored by the Everton's Dominic Nathaniel Calvert-Lewin.
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Aberdeen stepped up their efforts, but home substitute Emilio Zelaya added a second in the 86th minute to sink the men from the Granite City, with the club launching an investigation into disturbances in the stands after a video on social media appeared to show some visiting fans clashing with stewards.
Apollon Limassol beat Derek McInnes's side 2-0 on the night and 3-2 on aggregate but the match will be remembered for fighting in the stands and riot police separating supporters and stewards. "But we know we can and need to improve in our ball possession", he said.
"It's not easy to reach the group stages but we will do everything to get there".
"We always try to play offensive football, we don't come here to defend the result, it's not the way I like and it's not the best way. We need to create more than we did".