The Dons began well but fell behind to a goal from Jamie Murphy before Niall McGinn equalised in the final minute to earn a replay at Fir Park in their fourth round clash.
"I thought Motherwell should have been relieved not to be 3-0 down in the opening 10 minutes. We should have been ahead, but the usual quality of determination was apparent at the end," said Brown.
"We're a very good second half team because the team is very fit and their attitude is superb. With these qualities, I think we can go away from home and do even better."
Brown managed Murphy when he was Motherwell boss and his excellent strike came as no surprise to him.
"We lost a goal which came out of nothing," added Brown. "It was a superb goal from Jamie Murphy, and I'm not surprised by it because I know what he can do.
"You're always concerned when you lose a goal as late as that. I thought it would have been an injustice had we lost the game, and in the early part of the game we had to commend Lee Hollis, the Motherwell goalkeeper."
Meanwhile, Brown confirmed he could be boosted by the return of some of his eight injured first team players for the replay.
"We're missing eight players who quite conceivably could be in the team, and one or two have been outstanding throughout the season," he said.