Gary Harkins netted both goals for the Dark Blues, including a 77th minute winner, after Kris Boyd had equalised for the hosts a few minutes earlier.
Dundee's only previous win on opposition soil had come in early September and Brown, the interim manager, said: "They have been that way since I came in.
"We had hunger in the game against Motherwell last week but we gave away cheap goals. But that hunger is the dressing room.
"What surprises me is how they didn't have more points on the board. I don't know if I have had a reaction but they are fighting for each other.
"It's a great result. We have not had many wins this season but to come to Kilmarnock and win is fantastic."