The former Rangers defender took over from Barry Smith as a temporary measure in late February but the position is now his for keeps after signing a two-year deal. The 51-year-old has lost just one of his seven league games in charge as he bolstered the fragile side left in the wake of Smith's sacking.
A win for St Mirren at Hibernian on Saturday, however, will send Dundee back to the Irn-Bru First Division but Brown believes his side could yet be playing top-flight football next term. Asked if he thought he would be working in the second tier after the summer, Brown said: "No I don't."
He continued: "We are still fighting to stay in the Premier League. If we can take it another week and make a few people very nervous then great. People had us down a number of weeks ago. The players have shown the right attitude and managed to get results.
"We need to win and then look elsewhere and see what the result is. But we're not out of it."
With just four games left, a Saints win at Easter Road or anything but a victory for Dundee at home to Hearts and Brown's first act as permanent boss will be to pick up the pieces from a relegation.
But Dark Blues chief executive Scott Gardiner believes his new manager has merited his appointment.
"It was a straightforward decision in the end," he said.
"Results are everything in football and obviously John has got results. I'm not surprised by the impact he's made because I have appointed him but I am surprised at just how well he's done.
"We have lost just once in the league since he took over and if you had asked me before we gave him the job if losing one game out of seven would be acceptable, I'd have said that would be really something. But what John has done has been really something, hence why he's got the job."