With Celtic all but guaranteed to take top spot and the Champions League space that goes with it, the teams finishing in second and third places this year will qualify for the Europa League, along with the winners of the Scottish Cup.
Aberdeen exited that competition last week but Brown is confident Europe is still a possibility despite the disappointment of Saturday's 0-0 draw.
Brown said: "Everyone in the dressing room is enthusiastic and I'm still optimistic about a good finish to the season. We've got a dozen games left, and after we lost the first game of the season, harshly, to Celtic we went on an unbeaten run of 11 games.
"I think we can do the same again. There's enough in the dressing room to do that, and we can hopefully get more wins than draws this time.
"I was in the dressing room for quite a while, and there's a real determination to finish with a flourish. We can see that we can be the better team in games and we've just got to translate that into victories."
The former Scotland boss refused to comment on the second booking that saw Mark Reynolds sent off, the defender having been penalised for what referee John Beaton deemed a deliberate handball as he charged down Kenny McLean's 73rd-minute shot.
"You know I never make comment about that [referees], and I'm not going to change the habit."
St Mirren's in-form striker Esmael Goncalves failed to score for the first time in four games since joining the Buddies on loan from Rio Ave last month, but the striker had the best chance of the match with five minutes left.
Having been played in by Paul McGowan, the striker attempted to lift the ball over the onrushing Jamie Langfield, only for the goalkeeper to block well.
That was a chance would have left manager Danny Lennon disappointed his side did not earn a victory, but he pointed to the strides his side have made to be in a position where taking only a point from Pittodrie represents a disappointment.
"It shows you where the club has progressed from. We might still be at the bottom end of the table, but with the league so competitive, we find ourselves nine points away from second place. A couple of years ago, we were 36 or 37 points away from second.
"We also got another clean sheet, which is so important. Over the season, that has probably been our Achilles Heel."