Nottinghamshire Police launched an investigation following the incident at the City Ground.
The player, who was stood on the sidelines, was knocked to the ground as fans poured onto the pitch to celebrate Forest’s penalty shoot-out win.
Notts Police said it is continuing to liaise with both clubs as enquiries continue.
Chief Inspector Paul Hennessy, of Nottinghamshire Police, said: “A 31-year-old man has been arrested on suspicion of assault in connection with this incident. He remains in custody and will be questioned by officers.
“Our enquiries into the circumstances surrounding the incident are ongoing. We have spoken to the player concerned and we are working with both clubs who are supporting us with our investigation.”
Sheffield United boss Paul Heckingbottom revealed the incident left 36-year-old Sharp, who did not feature in the game because of an injury, needing stitches.
A Nottingham Forest club statement, released last night, said: “Nottingham Forest Football Club are appalled to learn that our former player, Billy Sharp, was assaulted leaving the pitch after tonight’s match at the City Ground.
"The club will work with the authorities to locate the individual in order they are held to account for their actions, which will include a life ban from Nottingham Forest.
"The club would also like to apologise to Billy personally and to Sheffield United Football Club.”
Forest won the tie 3-2 on penalties after the two legs finished 3-3 on aggregate.