Dundee looked set for their 20th defeat of their SPL campaign after Steven MacLean and Dave Mackay scored either side of John Baird's reply, but substitute Mark Stewart struck in the last minute to earn a point for Brown's 10-man team.
Despite currently sitting bottom of the SPL, interim boss Brown says his side will head into their William Hill Scottish Cup quarter-final against city rivals Dundee United on Sunday with a fearless attitude.
"We're not frightened of anybody. United have got a great record. They have got a good team with some excellent players," he said.
"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.
"I can't criticise the boys one bit for application and some of our football was excellent."