Tayside Police said the trouble was caused by a minority.
Officers said most of the capacity 4,100 support at Glebe Park conducted themselves "in an entirely appropriate way".
A police spokesman said: "The crowd were in good spirits as the game moved into extra time and the atmosphere was excellent.
"A minority tried to disrupt the game by setting off smoke bombs, with one person shouting sectarian remarks and another climbing on a floodlight structure.
"A 23-year-old male was arrested for a breach of the peace in relation to the floodlight incident and enquiries are ongoing in conjunction with Strathclyde Police to identify those responsible for the other offences.
"The actions of the minority should not overshadow the football game that was well attended, well supported, with fans from both sides conducting themselves in an entirely appropriate way."