Chief executive Stewart Regan said the Scottish Football Association was "appalled by the scenes of disorder and the contemptible behaviour" after the William Hill Scottish Cup final.
After Hibernian ended a 114-year wait to win the trophy with a 3-2 victory over Rangers, thousands of Hibs fans streamed on to the pitch at the final whistle at Hampden Park.
Mr Regan said that a full investigation, carried out along with Police Scotland, would begin.
He said: "The Scottish FA is appalled by the scenes of disorder on the field immediately after full-time and at the contemptible behaviour that ensued. What should have been an historic, memorable Scottish Cup final will now, sadly, be remembered for all the wrong reasons.
"Police Scotland and the stadium management initiated a Crisis Liaison Group as soon as the disorder began and we thank them for containing the situation as quickly as possible.
"The Scottish FA, along with Police Scotland and the stadium management have initiated a full investigation into the scenes that unfolded to understand how such a volume of supporters were able to enter the field."
Hibs pitchy. https://t.co/8jLObEQgmn— Limbs AOTS (@Limbs_AOTS) May 21, 2016
It is being reported that fighting broke out when some Rangers fans then got past stewards.
Speaking after the match, Hibernian boss Alan Stubbs told Sky Sports 1: ''I wish all the fans had left the pitch so we could celebrate it properly.
''We don't condone what has just happened, but that is 114 years of hurt.
Wow. Incredible rendition of Sunshine on Leith from the Hibs fans. Sky Sports 1. https://t.co/45a0SGb5EJ— Sky Sports Premier League (@SkySportsPL) May 21, 2016
"Everyone's safety is the most important thing. I can understand the euphoria that surrounded it. I don't condone it, but I understand it."
Police horses were eventually brought on to restore order as police and stewards struggled to cope.
Hibs won the match 3-2 with a David Gray goal in injury time.
Rangers said players and staff were assaulted by Hibs fans who invaded the pitch.
Hampden officials said the players would not return to the stand to collect their runners-up medals, receiving them in the changing room instead.
The club commended its own supporters for showing restraint under "severe provocation".
A statement said: "Rangers Football Club is shocked at the disgraceful behaviour of Hibs fans at full time.
"Rangers players and staff were assaulted by these fans who invaded the pitch and in the interests of their safety could not return to the pitch for the medal presentation.
"There can be no place for the violent behaviour witnessed at the end of the final and Rangers fully expect the Scottish Football Association and Police Scotland to launch an investigation to find out where security measures failed.
"The club would like to commend Rangers fans for the restraint the showed under severe provocation."
Police horses were last needed at the 1980 Scottish Cup final between Celtic and Rangers.
As Hibs fans returned to the stands part of the pitch was ripped up and the goal posts were broken by people climbing on them.