Police Scotland believe there could have been deaths at the Scottish Cup final had officers intervened to stop the pitch invasion at full-time.
The force said the disturbances which marred the fixture between Hibs and Rangers left officers and stewards with “no option” after supporters entered the field of play.
They added that any attempt to stop fans coming on to the pitch could have had a “catastrophic effect”.
The response came in a letter to a member of the public after an official complaint which raised concerns about the lack of policing at the match and failure to control the pitch invasion and subsequent disorder.
The five-page response said it would not have been feasible “to have hundreds, if not thousands of officers on standby”.
Earlier this month, a report commissioned by the Scottish Football Association suggested there should have been a reserve of police officers available to call on to deal with the disturbance.
But the author of the Police Scotland letter said: “In the face of thousands of people determined to enter the playing surface, the police and stewards had little or no option in preventing this.
“I can confirm that there were significant numbers of officers deployed to police this event and this was in line with our experience of previous matches . . . I would also point out that, in my opinion, had there been sufficient officers to deter those in the front rows from entering the pitch, any such action could have had a catastrophic effect had the many thousands of fans at the rear of the stadium surged forward.
“In such a scenario, a crushing situation would have been inevitable and could have resulted in serious injuries or fatalities.”
The SFA-commissioned report by Sheriff Principal Edward Bowen, published earlier this month, said the “vast majority” of fans who ran on to the field did so “in a spirit of jubilation”.
However, it said the Scottish Government should consider making it a criminal offence to run on to a football pitch.
A police spokeswoman said: “The opinion of the responding officer expressed in the correspondence, which deals with a quality of service complaint from a member of the public, seeks to provide additional context around the initial complaint which related to the pitch incursion and the way officers were deployed.”
Emergency services said 40 fans needed medical treatment after fighting broke out on the pitch, with one man suffering a punctured lung.
Hibs issued an apology following their historic 3-2 victory, with chairman Rod Petrie insisting: “I condemn and the club condemns all of the violence and unacceptable conduct perpetrated on the pitch and elsewhere and in the margins of the match at Hampden. There is no place for it in the game of football.
“Hibernian will do all that it can to support the criminal investigations to ensure that those who have transgressed face the consequences of their actions.”