Tony Pulis felt the timing of Everton's first two goals killed the game for West Brom after their 3-0 defeat at Goodison Park.
Everton regressed under the former Wigan Athletic manager but are now back to being the best of the rest under Koeman and harbour aspirations of regularly challenging the likes of Chelsea and Manchester City for silverware in the near future.
Everton's display emphasised their increased solidity in defence since Roberto Martinez's departure, while there were no signs of that negatively impacting their effectiveness going forward as they equalled last season's record despite having 10 more games to play.
A major priority will be discussions with both midfielder Ross Barkley and striker Romelu Lukaku about continuing their playing careers with Everton.
Everton's nine-match unbeaten Premier League run, stretching back to mid-December and garnering 21 points, was ended by a disappointing performance at Tottenham last weekend.
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It also showed that the Blues would only be seven goals worse off in the goal difference column without Romelu Lukaku's 18 league goals, highlighting this side may not be as dependant on the Belgian as first feared.
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Lukaku drove a free kick into the wall later in the half but the crowd's pleas for another goal were answered when Barkley popped up on the left side of the area, wrong-footed his marker and clipped an inch-perfect cross for Lukaku to place with a header into the corner to make it 3-0. Lukaku was played in and he whipped a right-foot shot just wide of the far post.
Everton claimed a 2-1 victory at the Hawthorns thanks to goals from Kevin Mirallas and Gareth Barry.
Lukaku finally got his name on the score sheet heading in Ross Barkley's cross from close range in the 83rd minute.