The departure of Moody, a close ally of Mackay, was made official in a brief club statement on Friday, which also confirmed little-known Kazakh Alisher Apsalyamov's appointment as acting head of recruitment.
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"I have every faith in Malky and his team to lead us through the challenges of the Premier League," the Malaysian told reporters.
"I have supported him in the past and will do so in the future for many years to come."
Many Cardiff supporters are understood to have been dismayed by news of Moody's departure, together with the hiring of Apsalyamov.
Tan stated: "I would say to all Cardiff City fans and everyone connected to the club: Let us look forward to the future and remain united in our support of the team."