The Dens Park outfit looked set for their 20th defeat of the Clydesdale Bank Premier League campaign after Steven MacLean and Dave Mackay had scored either side of John Baird's reply for Dundee to put them on the verge of victory.
But substitute Mark Stewart stuck in the last minute to give Brown's 10-man team - who lost Declan Gallacher to a straight red card for a last-man foul on MacLean - a morale-boosting point.
The SPL's basement side will face city rivals Dundee United in their William Hill Scottish Cup quarter-final at Dens on Sunday and the new manager claims his side will march into the tie fearless.
He said: "We're not frightened of anybody. United have got a great record. They have got a good team with some excellent players. But it's 11 v 11, isn't it?
"This result gives us a good platform. We have got a few problems like Gary Harkins being cup-tied.
"I've come here to assess all of the players from here until the end of the season and hopefully put my hat in the ring with all the other applications to get the job."
St Johnstone boss Steve Lomas, though, felt his side should have had the three points sewn up well before Stewart struck after appearing to be in an offside position.
Lomas said: "It's very much two points dropped. It can be a cruel game sometimes. We should have killed them off by scoring or through just being professional. Ultimately when you don't kill it off then the other team gets a chance.
"Dundee's goal was a little bit fortuitous but they got it and stuck it away so it's a missed opportunity. With the extra man you've got to kill the game off and we didn't do that. We've done it to other teams. There were some good saves from Rab Douglas but ultimately we should have been out of sight."