On Friday, Mackay was relieved of his duties after two-and-a-half years in charge of the Bluebirds.
The 41-year-old's exit had been widely predicted after Tan last week sent him an email demanding he resign or be sacked.
That ultimatum was briefly lifted when chairman Mehmet Dalman stated Mackay would be in charge for the "forseeable future", but the 3-0 Boxing Day defeat against Southampton proved the ex-Watford boss' last in charge.
A statement on the Cardiff website read: "The board of directors at Cardiff City Football Club have today relieved Malky Mackay of his duties.
"A new first-team manager will be appointed and announced in due course."
Mackay has often said he has felt sorry for the club's fans as Cardiff have found themselves in the headlines for the wrong reasons on several occasions.
But Tan laid the blame for that firmly at his former manager's door in a statement released to Sky Sports News.
"There has been a good deal of publicity generated by, and about, Mr Malky Mackay for the last few months," it read.
"Indeed, far too much dirty linen has been exposed to the public gaze but, I stress, not by me.
"Indeed, I have deliberately not responded to this, hoping that the club can be judged on its football rather than personalised arguments about who said what to whom.
"I have, however, regretfully concluded that it is no longer fair to the club, its players, its fans and the public more generally for this uncomfortable state of affairs to continue.
"Cardiff City Football Club means far too much to us all for it to be distracted by this."
Mackay had guided the Bluebirds to the League Cup final in 2012 and the Championship title in 2013. They lie 16th in the Premier League, one point above the relegation zone at the time of his departure.
He had always insisted he would not resign, and his removal had long appeared to be the only realistic outcome of his dispute with Tan which first came to light when head of recruitment Iain Moody was dismissed in October amid allegations of a transfer window overspend.
Mackay and Moody denied those claims and the Scot and Tan were at loggerheads again when the Bluebirds boss suggested he hoped to add three players to his squad in January.
Chief executive Simon Lim quickly responded with a statement which claimed Tan was "extremely upset" by Mackay's comments and that "not a single penny" would be available to spend next month.
That statement arrived the same day as Tan's ultimatum.
Even when Dalman offered Mackay the lifeline that he would remain in charge, it came with the caveat that there would need to be a reconciliation between owner and manager.
Mackay had twice requested a meeting with Tan and Dalman earlier this week in order to discuss their differences, but the Scot revealed the Malaysian had turned them down.
Talks had been due to be held on Saturday, when Cardiff face Sunderland, but in the end Mackay did not last long enough. His assistant David Kerslake and first-team coach Joe McBride will lead the team for that game.
Former Manchester United striker and Molde manager Ole Gunnar Solksjaer is among the favourties to replace Mackay, while Sven-Goran Eriksson and Turkish journeyman Yilmaz Vural have also been suggested as potential candidates.
And when responding to questions regarding Solksjaer's possible appointment, Cardiff's honorary life president and former owner Sam Hammam also voiced his displeasure at Mackay's tenure as manager.
He told BBC Wales: "He (Solskjaer) will have had very good basic grounding (at Molde) to what is needed now.
"We need a manager whose feet are on the ground and not just telling Vincent Tan, 'Give me your money,' as if all Vincent Tan is is a cash cow."