The Dons won this fourth round replay 2-1 at Fir Park, with Rory Fallon and Joe Shaughnessy netting for the visitors early in the second half, before Michael Higdon fired a last-gasp consolation from the penalty spot.
Manager Brown said: "We've now managed to turn it round against Motherwell. That's four games and it's two wins and two draws for us.
"Between the first and the second goal we had three really good chances and their goalkeeper Darren Randolph had two saves which were world class.
"We handled the situation exceptionally well, particularly the young boys out there. I thought they were superb.
"It was a big blow to us when we lost Russell Anderson on Monday. To come here without Russell and to get a victory against Motherwell is very good.
"I rate Motherwell highly. I rate Stuart McCall and I think they have a superb team."