Goals from substitute Lassad Nouioui and former Dons player Charlie Mulgrew secured the points for Neil Lennon's side but Brown felt his charges were a little unlucky.
"I don't think we did enough to win the game, but maybe to draw," he said.
"Other than the two goals, I don't think Jamie Langfield had a lot to do in terms of making saves or cutting out crosses. I may be a bit optimistic, but it looked like it would finish 0-0.
"It's a measure of champions that they win when they need to, but Celtic have lost to other teams in the SPL and I had hoped that there might be a chance for us to win."
Brown praised the fortitude of 18-year-old winger Ryan Fraser, who was substituted after 64 minutes having taken a few heavy knocks.
"He was winded. He was knocked about a bit," added Brown.
"I spoke to him when he came off and he was feeling a bit of pain, but he's a brave wee guy so hopefully he'll be back for next week.
"We had to replace Gavin Rae, who is very influential for us, at half-time with a hamstring, and we had started without Ryan Jack, Gary Naysmith and Rob Milsom.
"To lose another one at half-time didn't help us. It's not an excuse because we didn't do enough and didn't keep the ball the way we have been doing."