Abiola Dauda opened the scoring for the visitors in the 14th minute when he turned home a low cross.
They doubled their lead five minutes later through Denis Oliynyk, who fired in at the back post after his initial header had been blocked.
Kazaishvili scored their third of the evening with a close-range finish and Wallace soon added a fourth with a powerful strike from a tight angle.
Jeffrey Fortes pulled a goal back for the hosts with a composed finish into the corner, but Kazaishvili immediately restored his team's four-goal lead with a clipped effort over the keeper.
Ryan Koolwijk scored a second for Dordrecht with 11 minutes left to play, but in stoppage time Kazaishvili completed his hat-trick with the sixth goal of the evening for the visitors.