The defending champions rarely looked troubled, taking the lead after 21 minutes when Anwar El Ghazi managed to get behind the defence and square to Lasse Schone, who was left with a simple tap in.
Frank de Boer's men spurned several chances to extend their lead as the game wore on, but were given a boost on 63 minutes when Dordrecht's Josimar Lima was shown a second yellow card.
And Ajax wasted no time taking advantage, extending their lead just two minutes later as Schone doubled his tally with a curled free-kick.
That opened the floodgates and the home side had a third goal when Joel Veltman latched onto a Davy Klaassen ball to chip the ball over goalkeeper Filip Kurto in the 69th minute, before Lucas Andersen wrapped up the win with another strike six minutes later.
The result condemned Dordrecht to their fifth straight defeat, while Ajax sit four points adrift of leaders PSV at the top of the table after 11 matches.
PSV had to dig deep to grind out a 1-0 win at home to ADO Den Haag.
A bright opening spell from the hosts saw Memphis Depay cannon a volley off the crossbar before Jorrit Hendrix was given his marching orders for a second bookable offence after just half an hour.
But Philip Cocu's men fought back and took a 49th minute lead when the impressive Depay headed home Luciano Narsingh’s inch-perfect cross at the far post.
Despite their numerical advantage, Den Haag rarely troubled the hosts, who held on for their fourth consecutive league win.
Third-placed Feyenoord meanwhile stretched their winning streak to five games with a comfortable 2-0 win over PEC Zwolle.
A dominant performance from the hosts was not rewarded until four minutes into the second half when Jean-Paul Boetius and Terence Kongolo combined in the box to allow Sven van Beek to finish with aplomb.
The win was wrapped up 12 minutes later thanks to Boetius, who lashed home a vicious strike past keeper Warner Hahn.
Having suffered their first defeat of the season last Sunday, FC Twente returned to winning ways with a 2-0 victory at home to Heerenveen.
Rasmus Bengtsson and Youness Mokhtar were both on target for Twente, who climbed up to fourth in the standings.