The forward receives the ban for "abusive, insulting and threatening" language and behaviour, which has also landed him a £75,000 fine.
The ban applies to all domestic English games and includes one game already served, meaning Mitrovic will be forced to sit out the club's next seven fixtures.
Fulham manager Marco Silva has received a two-match ban and £20,000 fine after being sent off during the same game.
Fulham have also been fined £40,000 for failing to control their players in the ensuing scenes after the incident.
A statement from the FA read: "Fulham FC, Marco Silva and Aleksandar Mitrović have been sanctioned by an independent Regulatory Commission following a hearing in relation to their tie against Manchester United in The FA Cup on Sunday 19 March.
"We claimed the standard punishment which would otherwise apply to Aleksandar Mitrović for the sending-off offence of violent conduct that he committed towards the match referee around the 72nd minute was clearly insufficient.
"The forward denied this, but it was upheld by the Regulatory Commission, and a three-match ban was imposed. This is in addition to the three-match ban he had already received for the red card offence.
"The forward separately admitted that his behaviour and language after being sent off was improper, abusive, insulting and threatening, and the Regulatory Commission imposed an additional two-match ban and £75,000 fine.
In total, the Regulatory Commission has suspended Aleksandar Mitrović for eight matches. One of these has already been served, and therefore the forward will be unavailable for Fulham FC’s next seven matches."
Mitrovic will next be available to play against Southampton on 13th May.
