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.
Yet Tan moved to allay fears over Mackay's future following days of speculation, insisting the Scot - who guided Cardiff to promotion from the Championship last season - has his full supports
"I have every faith in Malky and his team to lead us through the challenges of the Premier League," said Tan.
"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 added: "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 pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
