The incident occurred during an FA Cup tie between Forest and Leicester at the City Ground.
A supporter ran onto the pitch after the home side had scored their third goal, in a 4-1 victory, and threw punches at players.
Cameron Toner, 19, from Whetstone, admitted to assaulting Nottingham Forest's Keinan Davis, Brennan Johnson and Djed Spence during an appearance at Nottingham Magistrates' Court on Thursday.
Toner was sentenced to four months at a young offenders' institute and will be released after two months on licence.
He was also given the maximum football banning order of ten years, as reported by The Athletic.
The 19-year-old took cocaine on the Friday before the match and then drank beer and spirits from 11am before the 4pm kick-off on Saturday, February 6.
Toner said: "I was under the influence of alcohol, having started drinking at 11am. I was not myself.
"I wish to apologise to Nottingham Forest, their players and supporters. I also wish to apologise to my friends, my family and all those connected with Leicester City.
"It was probably the adrenaline that kicked in."
Prosecutor Sanjay Jerath said: "I am sure Leicester City do not condone this behaviour. He [Toner] has tarnished their reputation."
Toner will also likely be subject to a lifetime ban from Leicester.
After the match, the Foxes said: After the match, a Leicester City spokesman said: "The club is appalled by such behaviour and offers its apologies to Nottingham Forest and its players whose safety was compromised.
"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."