There were several incidents at the game on August 4 this year, when fans reacted to an injury-time winner from Ibrox defender Conor Goldson.
In October, Police Scotland released images of 20 people they wanted to speak to in connection with the scenes, which were captured on TV cameras.
The roof of Rugby Park's disabled section collapsed as jubilant fans celebrated the winner, with one person injured.
A Police Scotland news statement read: "Police Scotland can confirm that 13 boys and men, aged 14 to 29, have now been arrested and charged in connection with various incidents at Rugby Park, Kilmarnock, on 4 August, 2019.
"Reports have been sent to the Procurator Fiscal and officers are following positive lines of enquiry to trace a number of other individuals.
"Officers would like to thank the public for their assistance with this investigation so far."