Ryan Conroy's deflected 30-yard free-kick earned the Dark Blues their third consecutive victory and put them eight points behind the Buddies with three Clydesdale Bank Premier League games remaining.
Dundee could, however, be relegated without kicking a ball as St Mirren face Hearts at Tynecastle next Saturday, 24 hours before Dundee host Aberdeen.
"I think you will be smelling it (fear) coming from them," said Brown. "They are going to Tynecastle and it's never an easy place to go.
"So unless they have Old Spice to cover it then I think it will be an interesting weekend.
"It was a great day, it takes it another week. For the fans today, it was fantastic.
"Hearts put in a great shift with 10 men and if they do that with 11 men against St Mirren, you never know, we might take it another week."
Nicky Riley will be suspended next week after he was sent off in the last minute and the influential Gary Harkins missed out through injury.
"We think Gary has a cracked rib," Brown said. "It was through a challenge with big Steve (Simonsen), the goalkeeper, on Thursday but that's what we do in training.
"It's contact, you can't take that out the game. Hopefully he might be there (next week) but I would think he might be out for the rest of the season."
Brown added: "Nicky will learn by his mistake. It was a silly thing to do but we are not going to criticise too much, there was a lot of emotion out there.
"It was handbags, I think he raised his hands into his face. The referee was right but I remember the days you used to give out right hooks and it was still okay."