British media reported this week that Mackay's position was under scrutiny after Moody, a key member of his backroom staff, was placed on leave, with some reports suggesting the manager had been asked to resign, something the club have denied.
"I have every faith in Malky and his team to lead us through the challenges of the Premier League," Tan said in a statement on the club's website (www.cardiffcityfc.co.uk) on Friday.
"I have supported him in the past and will do so in the future for many years to come," added the Malaysian businessman whose money helped the Welsh side reach the Premier League for the first time when they were promoted last season.
Moody, an important member of Mackay's staff since joining the Scot at Watford, has been replaced by Alisher Apsalyamov, a 23-year-old from Kazakhstan who has been appointed on an interim basis.
British media have reported that Apsalyamov is a friend of Tan's son and has no background in the game.
Tan said Apsalyamov would focus on gathering data on individual players.
"Ultimate recruitment decisions of course remain the domain of the manager and majority shareholder," he said.
Tan had previously upset Cardiff fans by changing the club colours from blue to red as well as altering the team's crest, but he has brought success on the pitch.
He invested more than 30 million pounds ($47.84 million)on new players during the close season and Cardiff are 14th in the standings with eight points from seven matches.
"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," he added.
On Thursday, the Cardiff City Supporters' Trust issued a statement backing Mackay and calling for clarification of the situation, saying they feared developments at the club would see the manager leave for another Premier League team.
($1 = 0.6271 British pounds)