The ban was handed out by the Ulster Council earlier this month but news of it only emerged on Friday morning.
This followed Barton's decision to withdraw an appeal which had been scheduled for Thursday night.
Barton was among a number of Derry and Tyrone personnel sent off in the McKenna Cup decider on 23 January.
The second-half incident in which Barton became embroiled in followed  a dangerous sliding challenge by Derry forward Daniel McKinless on Tyrone defender Ronan McNamee.
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The Tyrone man reacted by pushing over McKinless, and appearing to make contact with the Derry man's throat in the process.
As other players and Barton joined the fray, Tyrone goalkeeper Niall Morgan attempted to pull up McKinless up off the ground and the Derry boss then pushed the Red Hand keeper before, in turn, being jostled by Cathal McCarron.
After order was restored by Cavan referee Noel Mooney, McKinless and McNamee were both handed yellow cards which resulted in the Tyrone man being sent off as he had been cautioned earlier.
Derry boss Barton was also ordered to the stand while McCarron received a straight red card for his part in the melee.
McCarron is understood to have been handed a one-match ban for his part in the melee which will apply in next year's Dr McKenna Cup.
Barton's ban means he will not be allowed to sit in the Derry dug-out in their five upcoming Football League Division 2 matches, beginning with Sunday's home game against Galway.
However, Barton will be able to continue to train the squad and attend the games.
Derry football manager Damian Barton has been given an eight-week ban as a result of his involvement in an on-field melee in the McKenna Cup final.