Advocaat is back coaching in the Dutch league after 14 years with a mission to win the title and take PSV back to the Champions League, which they have not graced for four seasons.
PSV dominated from the start and took a 2-0 lead after just 11 minutes with midfielder Van Bommel, who last played for the club in 2005 before joining Barcelona, driving them forward.
Sweden forward Ola Toivonen netted at the front post from a Dries Mertens corner after three minutes for the Dutch Cup winners and Jeremain Lens doubled their lead after recovering the ball up in midfield and finishing with a superb chip.
Ajax defender Toby Alderweireld was the most dangerous player for the injury-hit champions, whose average age was less than 23. He had a header cleared off the line by Van Bommel early on but pulled a goal back on the stroke of half-time.
Eight minutes after the break PSV left back Jetro Willems' fine individual run took him past three players before he set up Toivonen to restore a two-goal lead.
Marcelo's own goal 15 minutes from time raised Ajax's hopes of drawing level but PSV substitute Georginio Wijnaldum wrapped up the win when he ran onto a Lens' pass to score in added time.
