The former Sunderland boss, 69, is starting his third spell as national team boss.
Galatasaray midfielder Wesley Sneijder scored on the night he won a record 131st cap, while Tottenham striker Vincent Janssen added a late penalty.
The Dutch are still third in Group A.
Liverpool midfielder Georginio Wijnaldum, Bayern Munich forward Arjen Robben and Spartak Moscow’s Quincy Promes also found the net.
Only the winners of each group are guaranteed a place at next summer’s finals in Russia, with the eight best second-placed sides competing in the play-offs.
Sweden are the new leaders in Group A after an injury-time winner from Ola Toivonen saw them beat France 2-1.
Arsenal striker Olivier Giroud scored a stunning left-footed strike from 20 yards to put the previous leaders ahead but Jimmy Durmaz equalised and former Sunderland forward Toivonen scored in the 93rd minute from 50 yards after a poor clearance from Tottenham goalkeeper Hugo Lloris.
Sweden have not qualified for the finals since 2006.