The supporter was restrained by stewards and then escorted away from the City Ground Stadium by police.
It was the first meeting between the East Midlands' clubs since both were in the Championship in 2014.
Leicester City say they are "appalled".
Nottinghamshire Police chief inspector Neil Williams told BBC Sport: "We have arrested a man in connection with an incident that took place during the first half of the Nottingham Forest and Leicester City FA Cup match.
"Investigations are in the very early stages and our enquiries into the circumstances surrounding the incident are ongoing."
In response to the incident, Leicester have offered their apologies to their hosts.
"The club is appalled by such behaviour and offers its apologies to Nottingham Forest and its players whose safety was compromised," said the Foxes in a statement.
"We applaud the swift action taken by City Ground stewards in apprehending the individual involved, who will be subject to a lifetime ban from all Leicester City fixtures at home and away.
"We will offer every support to the relevant authorities in pursuing further appropriate action."
Leicester boss Brendan Rodgers said: "It's behaviour we do not want to see.
"The field has to be a safe environment, we must always retain the respect in football and unfortunately the guy has come onto the pitch.
"The rules of society have changed - there's no respect. People feel entitled to think they can do these things. We can't have that lack of respect."
The incident followed a plea by Nottinghamshire Police earlier on Sunday, which encouraged supporters not to "allow their emotions to get the better of them and spoil the day for others".
Leicester, who won the FA Cup for the first time in their history last season, were backed by around 4,000 supporters.
Police also responded to disorder in the city centre before kick-off.
Chief inspector Williams added: "Officers were quick to attend reports of a disturbance that broke out in Chapel Bar, in the city centre, at around 12:15 GMT today following reports of windows being damaged.
"Three people suffered very minor injuries during the incident before the fans were quickly moved on by officers in the area.
"Police were also quickly on the scene following reports of a fight and windows being smashed in Byard Lane, in the city centre, at around 15:20 this afternoon.
"We take any reports of violence extremely seriously and anyone found to be breaking the law and identified as being part of these incidents will be arrested and dealt with."