Gavin Rae and Scott Vernon netted as Brown's men booked a place in the last 16 of the Scottish Communities League Cup.
"There was a resilience about the Aberdeen team, which last season we didn't quite have," Brown told BBC Radio Scotland.
"We've got it this season. We're not great but we're improving."
Morton frustrated Aberdeen and created a few chances of their owns throughout a goalless 90 minutes at Cappielow.
But the Dons' two goals in the final 11 minutes of extra-time ensured that there was no need for penalties, with Aberdeen having been beaten in a shoot-out against East Fife in last season's League Cup.
"We've got players who will dig out a result not playing as well as we can play," Brown said.
"When we're not playing well and we manage to win, that shows the determination within the group and we've got characters there, guys who are winners.
"I'm absolutely delighted to win because it's a very tricky venue to come to. We've had difficulty before against teams from outwith the SPL and the club's been embarrassed in the past.
"Even although it took us to extra-time, I'm absolutely delighted that we've managed win.
"I thought Morton played exceptionally well. They made it very difficult and it took a long time for us to get the elusive goal.
"Having said that, Gavin Rae might have had four goals. He was in great scoring positions. It could've been a bigger victory but, believe me, I'm very pleased with the victory."
Morton boss Allan Moore praised the First Division side's endeavour.
"We defended valiantly," he said. "We can be proud of the way the boys battled away.
"Against Premier League teams you've got to take your opportunities and we probably had two or three.
"We're hoping that performance will give the boys heart."
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