Greg Binnie, 19, admitted a charge under the Offensive Behaviour Act following disorder after Saturday's game at Hampden, which Hibs won 3-2.
At Glasgow Sheriff Court, sentence on Binnie, from Ratho, near Edinburgh, was deferred and he was granted bail.
The Crown has requested a football banning order also be considered.
The court heard that after the final whistle Binnie had climbed over fencing onto the field of play.
The charge also states he engaged in behaviour likely to incite public disorder.
His solicitor told the court the incident had led to threats being made to Binnie and his family.
He said Binnie, who has no previous convictions, was deeply ashamed.
Meanwhile another teenager appeared on petition at the same court accused of assaulting Rangers players Lee Wallace and Jason Holt.
Dale Pryde, from Edinburgh, is accused of running towards and attempting to punch Wallace on the head at Hampden Park, and attempting to punch Holt.
Mr Pryde is also accused of breaching the peace by running on to the pitch towards Rangers supporters and brandishing a chair.
He made no plea or declaration and was released on bail.
He is expected to appear again at a later date.
A man has admitted running towards Rangers goalkeeper Wes Foderingham and gesticulating in an offensive manner after the Scottish Cup Final.