It took Ajax 22 minutes to open the scoring against opposition who have not won since the opening day of the season, as Schone met Anwar El Ghazi's ball to net his fifth goal of the campaign.
Second-from-bottom Dordrecht held firm until the break, though, and it was not until Lima picked up his second bookable offence that Frank de Boer's side began to firmly establish their dominance.
Only three minutes after gaining a man advantage, Ajax increased their advantage through Schone again.
Davy Klaassen's set-piece delivery set up unlikely goalscorer Veltman for the home side's third goal on 69 minutes, and the same man provided Andersen with the chance to complete the rout on 75 minutes.
The result lifts Ajax to within one point of PSV Eindhoven at the top of the Eredivisie table, while Dordrecht remain 17th after a fourth successive loss.