The club, whose manager Alan Stubbs quit last night for a new job with Rotherham, said the sanctions against fans identified as being involved in incidents at Hampden included life bans and indefinite bans.
But it would not be drawn on how many supporters had been banned or what precisely they had been banned for.
There were clashes between rival fans after thousands of Hibs supporters streamed on to the pitch to celebrate their 3-2 victory over Rangers, Hibs’ first Scottish Cup win for 114 years.
Rangers claimed some of their staff and players were assaulted and Hibs vowed to co-operate with police investigations and an independent commission set up by the Scottish Football Association.
Hibs chief executive Leeann Dempster said: “While Police Scotland will continue with their own inquiries, we will continue to review what happened and take the action we think appropriate to protect the club’s good name. We will also co-operate fully with the Independent Commission established by the SFA. We have initiated contact with the supporters informing them of our decision. We will continue to identify and impose appropriate sanctions against others involved in unacceptable behaviour.”
Mike Riley, chairman of Hibs Supporters Club southern branch, backed bans on any supporters involved in violence.
He said: “Any violent acts should get the stiffest of punishments. I think everyone feels that. I was part of the thousands that were there, shouting to get the fans off the park.
“Everyone who was left in the stands was clapping ‘Off, off, off’ and shouting to get them off the park.
“Some people were just there just out of exuberance – it was 114 years. A lot just went with the crowd, a lot possibly had to go because there was a surge. But where there was violence the most severe punishment is appropriate. I’m sure Hibs will deal with them correctly.”
After the match Hibs chairman Rod Petrie condemned violence and unacceptable conduct, which he said had “no place in the game of football”.
Police said 14 people had been arrested over incidents linked to the match. Officers are continuing to study video footage from the day and more arrests are expected.