The Scottish coach is said to have been furious at the club's decision to put Moody, Mackay's right-hand man, on initial gardening leave.
It has since been confirmed by the Premier League outfit that Moody has permanently left with the club's "thanks", and 23-year-old Alisher Apsalyamov has been placed in the role on an interim basis.
Tan told the club's official website: "I have every faith in Malky and his team to lead us through the challenges of the Premier League. I have supported him in the past and will do so in the future for many years to come.
"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."
The club have reiterated that Mackay and the club's majority shareholder have final say on player recruitment.