A total of four supporters were arrested both before and during the fourth round Scottish Cup tie on a number of football related and public order offences.
The post-match investigation will focus on the use of pyrotechnic devices, smoke bombs and fire crackers which were set off within the terraces as well as being thrown onto the field of play.
Police will also be investigating outbreaks of sectarian singing and chanting which is reported to have taken place.
Match commander Superintendent Steven Lowther said: “The behaviour of a minority on the day was unacceptable.
“What was supposed to be an enjoyable event for all was tarnished by a group of individuals who were intent on singing sectarian songs as well as letting off smoke bombs and pyrotechnics within the crowd.”
He added: “It is evident that those involved had no concerns for either the safety of others or how their behaviour spoiled what should otherwise have been an event where the post-match discussion should have centred on football and not their antics.
“We will work with partners and both clubs to try and identify those involved to ensure positive action is taken to address the scenes which were witnessed at Stair Park.”