Smith described the split, saying: "We decided we were going to separate for a period of time and you go figure out how to make yourself happy and I will go figure out how to make myself happy."
Pinkett Smith added: "From there, as time went on, I got into a different kind of entanglement with August."
The 48-year-old actress also said she wanted "to clear up" reports that Smith, 51, had given her "permission" to stray.
In a radio interview with The Breakfast Club, Alsina had confirmed that he and Pinkett Smith dated during the marriage and said it was with the full-knowledge and "blessing" of Smith - leading to speculation the couple were in an open marriage.
Pinkett Smith clarified: "The only person who can give permission in that particular circumstance is myself".