Having lost 2-0 to the Kazakhstan in the first leg, Karagandy frequently dismissed the Scottish champions' chances of progression to the group stages prior to their meeting at Celtic Park on Wednesday.
However, the Bhoys managed to defy the odds and advance to the next round after a 3-0 win which Brown feels was fully warranted.
"That was sweet," Brown told reporters. "They came here saying they expected to score two or three but everybody underestimated us and we showed how good a team we really are. We were angry about everything last week.
"The only shot they had on goal tonight came from a long throw-in when they hit the bar. That's all they have: long throws.
"I got booked for celebrating the way they celebrated at the end of the first game. They were in everybody's faces. I know we are a big club and they were delighted to beat us but you have to show a bit of respect and they got what they deserved."
The draw for the group stages will be made on Thursday afternoon.