Saints claimed a first win in seven games thanks to a double from Jay Rodriguez and a Rickie Lambert strike in the space of 13 first-half minutes.
Peter Whittingham missed a great early chance for the Bluebirds but, aside from a brief rally early in the second half, they were outplayed by their visitors.
The on-field performance added to the sense of off-field discontent caused by the civil war between Mackay and Tan which has put the manager's position in jeopardy.
The lingering uncertainty over Mackay's future, despite a statement from chairman Mehmet Dalman that he would be in charge for the "foreseeable future", fuelled a pre-match protest where a group of supporters voiced their disapproval of Tan's running of the club.
Chants of 'Don't sack Malky Mackay' and 'We'll always be blue' also rang around the stadium during the game.
But the woeful nature of the performance was such that by the full-time whistle not all of the boos from the remaining home fans were for the Malaysian owner, who himself left his seat five minutes before the end.
The result only heaps further scrutiny on Mackay and his team, but the Scot was adamant his predicament could not be used as an excuse.
"I'm certainly not going to make any excuses at all as far as that is concerned," he said. "People can draw their own conclusions. Myself, the staff and the players are fully focused on what we do on a Saturday or any given game day."
But the Scot also insisted he does not know when he will hold crunch talks with Tan and Dalman which could decide his future.
Mackay revealed earlier this week that Tan had rejected two requests for the duo and Dalman to meet in an attempt to reconcile their differences.
Press Association Sport understands the meeting could take place on Saturday, when the Bluebirds are due to face Sunderland.
But when asked if a date had been set, Mackay said: "Not as far as I'm aware. I've not been told of any meeting. I don't usually talk to him (Tan) on a game day. That's not something that normally happens. I usually speak to other people.
"I was informed we would try to have a meeting but there has been nothing to this point."
While Lambert and Rodriguez got the goals for the visitors, it was Adam Lallana who proved the stand-out player, with his movement and class on the ball proving too much for Cardiff to handle.
It caps a memorable couple of days for the midfielder, who got married on Christmas Eve.
The 25-year-old has been linked with the likes of Manchester United, but manager Mauricio Pochettino is confident he can keep him on the south coast.
The Argentinian said: "I think Adam and other players are going to stay with us. There is no pressure to sell them, Adam has said he is happy at Southampton and Southampton have no plans to sell him."
He added of Lallana's chance of going to the World Cup with England: "I think it is too soon to be talking about the World Cup but it is clear players like Adam are candidates for the World Cup."