A brace from Jay Rodriguez and a further goal from Rickie Lambert inside the opening half hour condemned Cardiff to a second consecutive defeat and piled more pressure onto Mackay, whose future has come under scrutiny following a public fall-out with owner Vincent Tan.
After the match, Mackay reiterated that he will remain in charge of the team until told otherwise.
"I have belief in my ability to be the manager here and I'm standing strong to make sure if it's anything to do with me and my staff we'll make sure we stay in this league," he said.
"It's certainly very important that we stop washing our dirty linen in public.
"Our fans have been seen in a great light this season. The atmosphere; the friendliness; the way they're conducting themselves around the country is something they should be proud of and it's not great when we're on the front pages as opposed to the back pages.
"I truly hope for our fans' sake that they're able to enjoy the rest of this Premier League season."
Mackay refused to use Cardiff's off-field problems as an excuse for defeat against a Southampton side that had previously gone six league matches without a win.
"I'm not making any excuses," he said.
"People can draw their own conclusions but myself, the staff and the players are fully focused on what we do.
"We gave away three poor goals and we can't do that against a good team. It was unlike us. We can't give ourselves that mountain to climb but we can't give cheap goals away."
