The Toffees were defeated by Tottenham last time out, ending a nine-game unbeaten top-flight run, but offered a spirited response against a side who started the day only four points behind them.
Kevin Mirallas, Morgan Schneiderlin and Romelu Lukaku - a former West Brom loanee - were on target at Goodison Park, leaving manager Koeman delighted.
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“It was an outstanding team performance. In the second half, we controlled it. Our defensive organisation from set-plays was outstanding,” said the Dutchman in quotes reported by BBC Sport.
“We will always score because we have those types of players. Normally Romelu Lukaku will score in every game. But you need to fight, defend and run. The pressing in midfield was outstanding.
“Our target is to play next season in Europe. We know maybe seventh position will bring it.”
Ross Barkley was singled out for praise after playing a key role in two of his side’s goals.
“He’s improving,” added Koeman. “He has a lot more freedom now in the team. He’s a fantastic player. [He was] outstanding.”
Baggies boss Tony Pulis also identified Barkley as having been influential and admitted Everton were simply the better side.
“Conceding the two goals before half-time killed us. Before that we were in the game,” he said. “Romelu Lukaku and Ross Barkley were outstanding.
“We created some opportunities. Our biggest problem in the last couple of weeks has been the little bit of quality we need with a final pass or shot or cross. Everton had that.
“Everton are the best team outside the top six. They’ve got real quality.”
