The south Wales side also announced that 23-year-old Kazakhstani Alisher Apsalyamov has been appointed as the acting head of recruitment on an interim basis.
The loss of Moody had placed Mackay's position under scrutiny but Press Association Sport understands the Scot has no intention of resigning, and the club have denied reports Mackay was asked to leave his post.
Tan said in a statement released on Friday afternoon: "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."