For 82 minutes, Everton were very firmly in the match despite Antonio Valencia's cracking opener.
Anthony Martial's injury-time penalty completed the rout, meaning Everton has conceded 10 goals without scoring in eight days.
Koeman's side are now third from bottom in the Premier League table and have won only one game, conceding nine goals in their three successive league defeats.
Despite Ronald Koeman making nine senior signings to bolster his squad including the return of Wayne Rooney and the club-record signing of Gylfi Sigurdsson, the Toffees have struggled to hit top form and their woes were compounded in midweek when they lost 3-0 away at Atalanta in the Europa League. Frank de Boer's sacking after just four games by Crystal Palace underlined the need for instant success in the Premier League and bookies' odds on Koeman becoming the next manager to be sacked tumbled over the weekend.
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Given the former England captain's recent seasons with Manchester United, it was unrealistic to believe he'd slot in at Goodison Park and replace Lukaku's goals, despite a promising start.
Whilst speaking about the situation, Koeman said, "I read the programme and my colleague said they are spending £140m and need to go for the top four". Its next league games are at home to Bournemouth and Burnley.
"We grew in the game and in confidence and not many teams will get two big chances like we got today".
Yannick Bolasie is confident that Everton will bounce back from their disappointing 4-0 loss against Manchester United on Sunday.