The Dee's 2-1 win in Paisley over St Mirren ensures their bid to beat the Clydesdale Bank Premier League drop will linger on until next week's home fixture with Hearts.
But the late winner struck by Carl Finnigan - back in action after almost a year out injured - will do wonders for the squad's morale and raises hopes they can pull of an unlikely escape act.
Brown has now lost just two of his eight games in charge since replacing former manager Barry Smith on a temporary basis and, when asked if he'd like the post permanently, Brown replied: "Absolutely. Dundee is a big club. I had a great playing career here in my early years and the Dundee fans have always given us great backing. So I would love the opportunity."
The Dark Blues travelled to Renfrewshire knowing anything short of a win would relegate them from the top flight after just one season but they have now closed the gap on Saints to 10 points with four games remaining.
Jim McAlister put Dundee ahead five minutes before the break after Marc McAusland's slip but Steven Thompson levelled seven minutes into the second period when he thudded home a header at the back post.
But after losing skipper Jim Goodwin following two needless bookings, Saints found it harder to repel their desperate visitors, who finally struck eight minutes before the end with Finnigan heading in off the underside of the bar.
Saints manager Danny Lennon admitted his team's chances had been hurt by the red card dished out by referee Craig Thompson to Goodwin, and he said: "Jim is an important and experienced player for us but he put himself and the team in a difficult position and Craig ran out of patience with him."
The Buddies boss was impressed by their opponents' fight and hopes they can replicate it next week and end their own relegation fears before panic sets in.
"Dundee brought a fighting spirit and desire and that's something I'm hoping my players can learn from in the last four games," he added. "When we bring our industry, it brings out our strengths. But when we don't, we are very ordinary. In fact we were below ordinary."