Their form since the former Dundee and Rangers defender replaced Barry Smith has narrowed the gap on the Scottish Communities League Cup winners to eight points.
"There's three games to go, nine points and St Mirren have downed tools since the League Cup final and it's very difficult to pick that up," suggested Brown.
Since beating Hearts 3-2 at Hampden Park, Danny Lennon's side have had four draws and one defeat - to Brown's Dundee - and on Saturday face an Edinburgh side who will be looking to avenge that loss.
Chasing a fourth straight win against Aberdeen on Sunday, Dundee know they will already have been relegated if St Mirren beat Hearts on Saturday.
But Brown said: "Tynecastle is a difficult place to go to, especially against a team they beat in the final and I'm sure Gary Locke will be telling them it was a final they could have won."
Asked what he had changed to revive Dundee's fortunes, Brown told BBC Scotland: "I would expect players to be totally committed for the whole game and maintain concentration levels.
Interview - Dundee manager John Brown
"I think in some of the games I watched during the season there was a slackness on the counter-attack when Dundee lost quite a few goals - and slack goals at set-pieces - and we've tried to tighten up.
"But we've had a go at teams and that's the one thing I tell the players - to go and enjoy their football.
"When I came in, we were 15 adrift and I just said 'let's go and enjoy the football, work hard and see where it takes us'.
"It's an award for the players, for the reaction they've given me and proving a few doubters wrong and showing they are not First Division players."
Brown's appointment had not been universally popular with Dundee fans, some of whom questioned the suitability of an ex-player whose only management experience was a year with Clyde.
"There's was a minority that stir things up and they are on social media sites but they don't faze me," he said.
Even if Dundee fail in their bid to escape relegation to Division One, Brown has set his sights on eventually overtaking neighbours United, who this season secured a top-six finish.
Asked what his ambitions are for Dundee, he said: "Build it up to be the best club in Dundee and we'll take it from there."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.