Swede Ola Toivonen headed home the winner four minutes from time for second-placed PSV after Alkmaar were reduced to 10 men following a second booking for Mounir El Hamdaoui in the 71st minute.
Advocaat, also coach of the Belgium national team, accepted Alkmaar's offer earlier this week to succeed Ronald Koeman who was sacked by the Dutch champions last Saturday.
For Advocaat it was his first match in the Dutch league since May 1998 when he left PSV to join Rangers.
Leaders Twente Enschede, on 47 points, came from behind to beat NAC Breda 3-1 and maintain their two-point lead.
Blaise Nkufo missed a penalty for Steve McClaren's men after 14 minutes and Breda's Edwin de Graaf opened the scoring minutes before the interval with a shot from a tight angle.
Miroslav Stoch, on loan from Chelsea, converted a 58th minute penalty to equalise and added a second nine minutes later, while Bryan Ruiz lifted his tally to 13 for the season to seal Twente's 11th straight win.
Ajax Amsterdam, 3-0 winners against NEC Nijmegen on Friday, are third with 38 points, four more then Feyenoord who beat Heerenveen 2-0.
Kevin Hofland opened the scoring from close range midway through the second half and Jon Dahl Tomasson settled the match with a superb lob eight minutes from time to earn Feyenoord their first win at Heerenveen in 11 years.
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