The third inquiry, conducted by Katharine Newton, found that Sampson did tell Aluko to be careful her Nigerian relatives did not bring the Ebola virus to Wembley and also upset Spence, a mixed-race player, by asking her how many times she had been arrested.
"On behalf of the Football Association I would like to sincerely apologise to Eniola Aluko and Drew Spence," FA chief executive Martin Glenn said in a statement.
"Based on new evidence submitted to independent barrister Katharine Newton, she has now found that they were both subject to discriminatory remarks made by an FA employee. This is not acceptable.
"In her final report Katharine Newton concluded that on two separate occasions Mark Sampson made ill-judged attempts at humour, which as a matter of law were discriminatory on grounds of race within the meaning of the Equality Act 2010. Katharine Newton did however conclude that Mark Sampson was not racist.
"She also concluded that there was no evidence to support the allegations that Eniola Aluko was subjected to 'a course of bullying and discriminatory conduct' by Mark Sampson.
"Our ambition has always been to find the truth and take swift and appropriate action if needed. It was our decision to have the original, second and final investigation to ensure that due diligence was taken. It is regrettable that Eniola did not participate in the first external investigation as this would have enabled Katharine Newton to conduct and complete her investigation sooner. We will fully support the recommendations from the report."
Sampson was sacked as England boss last month after evidence of "inappropriate and unacceptable" behaviour with female players in a previous role.