The incident was alerted to the Football Association following Millwall's 2-1 win over the former Premier League side at The Den last month.
A statement released by Millwall said: "Having completed our investigations into the incident, we have identified and interviewed a 13-year-old boy who admitted a verbal exchange with Marvin Sordell in which abuse was used.
"The individual has been banned from Millwall matches for the foreseeable future, but as a club we also felt, given the boy's age and background, we had a duty to play our part in attempting to educate and rehabilitate him.
"Accordingly, we have offered to put him through one of our education programmes, run by Millwall for All in the hope that we can change his outlook on equality, racism and life in general."
The club also confirmed that the teenager had apologised to Sordell via a letter, which the 21-year-old has accepted.