The Scottish manager was sacked last week following a 10-month stint that had seen United lose 15 matches in all competitions.
However, many pundits believe that one of Moyes's best pieces of work was securing the services of Mata from Chelsea in January for a club-record £37m.
"Obviously, I don't want to forget that, since the beginning of the week, David Moyes is no longer our manager. I wish him and his technical staff the best of luck in the future, and I do thank him for his continuous interest in me and his importance during my transfer to become a Manchester United player," the Spaniard wrote on his blog page.
Mata came off the bench to score twice as United defeated Norwich City 4-0 at the weekend.