Scottish Football Association chief executive Stewart Regan has condemned "the inflammatory and irresponsible behaviour'' in Celtic's 1-0 win over Rangers in Wednesday night's Scottish Cup fifth-round replay at Celtic Park, and confirmed an investigation has been launched into all incidents.
Celtic boss Neil Lennon and Rangers assistant manager Ally McCoist had to be separated after the game, in which the home side won 1-0 with a Mark Wilson goal.
Rangers duo Steven Whittaker and Madjid Bougherra were sent off during the match, both for two yellow cards, and after the final whistle team-mate El-Hadji Diouf also picked up a red card for a second booking after approaching referee Calum Murray.
Regan said: "The Scottish FA categorically condemns the inflammatory and irresponsible behaviour throughout last night's Scottish Cup replay between Celtic and Rangers at Celtic Park. As Chief Executive of this organisation, I was both saddened and deeply embarrassed to witness the scenes that unfolded during what is supposed to be Scottish football's flagship fixture: these images were broadcast around the world and shows our game in a poor light.
"I acknowledge the pressures of expectation on both clubs but last night's behaviour crossed the boundaries of acceptable conduct at a football match. We have already launched an investigation into all incidents that occurred and will do everything in our power to ensure there is no repeat.
"The events at Celtic Park, however, run deeper than the Scottish FA's Disciplinary Procedures. The unedifying sight of two of the country's most recognisable and respected coaches engaged in an angry confrontation was not only unsavoury but exacerbated an already incendiary atmosphere inside the stadium and throughout the West of Scotland.
"The clubs have a duty of care to ensure that the image and integrity of the game is upheld at all times. This was not adhered to last night. This week, Strathclyde Police reiterated their concerns over the heightening violence and public disorder around Old Firm derbies. It is incumbent on Rangers and Celtic to ensure a far more responsible level of behaviour.
"In an age of austerity and financial hardship, football must try harder than ever in this country to restore its image as the national sport, extolling pride and passion. It fails in this regard. Last night's actions - which culminated in three red cards and 13 cautions in total - also re-emphasised the ongoing lack of respect for our match officials.''
The Scottish Police Federation has called for an end to Old Firm "madness''. SPF chairman Les Gray said police no longer had the budget to deal with match-related unrest, and he called for the derby to be played behind closed doors or banned altogether.
Mr Gray said: "We simply don't have the money and resources to do this. Everyone involved needs to sit down and look at this. Something has to give. This madness cannot go on.''
Strathclyde Police said arrests were for a variety of sectarian, racial and breach of the peace offences. Police had warned they would crack down on drink-fuelled violence linked to the match after trouble flared in the wake of the last Old Firm game.
On February 20, more than 229 people were arrested in the force area and in some cases prisoners were said to have been driven 50 miles as police cells filled up.
Mr Gray said consideration should be given to playing the matches behind closed doors or restricting the television coverage.
Speaking after the latest clash, he said: "What happens on the pitch is reproduced throughout Scotland, on the streets, in pubs, in homes. You cannot justify it. It can't keep on going.''
Justice Secretary Kenny MacAskill said the violence has more to do with alcohol and is less about football. He said: "There is a deep-rooted and unacceptable social issue involved - which has a great deal to do with Scotland's damaging relationship with alcohol, and little to do with football. The issue is far wider than clubs and the game. It is about a culture of violence fuelled by alcohol.
"By all means drink responsibly and safely but do not get drunk, get violent, assault your partner or anyone else - the message is that any such conduct is utterly shameful and unacceptable.''