The 56-year-old informed the Dutch outfit's board on Wednesday that he will be leaving to take up a new role with the Blues on June 1.
Everton have announced the appointment of Marcel Brands as the club's new Director of Football.
Brands said: "The fantastic passion of the fans for their club, the magnificent history of the club over the years and the ambition and goal we have as people tasked with driving the club forward all mean that we need to be competing for honours against the Premier League elite".
He added: "It is no straightforward challenge but this must be our motivation each and every day".
Allardyce was an unpopular choice with fans from the start but, having failed to lure Silva from Watford to replace Koeman, Moshiri took a pragmatic decision and went with chairman Bill Kenwright's recommendation to steer the club away from any potential relegation trouble.
Given their latest acquisition's reputation as one of the best talent spotters in Dutch football, Everton fans have already begun to speculate on who could arrive at the club this summer on Brands' recommendation, with talented playmaker Hakim Ziyech and Mexico worldwide Hirving Lozano already popular shouts.
'It is just a problem akin to this which may have persuaded me to depart PSV the place I'm indebted to the many individuals whose collective effort led to some nice achievements. I now get the chance to do something in my own way.
Everton Director, Professor Denise Barrett-Baxendale, said: "I'm very much looking forward to working closely with Marcel whose experience in the areas of player recruitment and development pathways for young players has forged an outstanding reputation in European football. Now we will look to build something really strong and lasting here at Everton".
But his role in a spending spree which exceeded £100 million last summer and yielded little successes ultimately played a part in his Everton downfall.
Walsh was headhunted from Leicester City in 2016 having been credited for overseeing their triumphant title season.