The Motherwell outfit surged into a 2-0 lead within 32 minutes thanks to skipper Ross Lindsay’s curler and Jamie Watson’s cool finish.
But their prospects were hit just five minutes later when Duncan was dismissed after a late tackle.
Despite their numerical disadvantage, Braves added their third goal within two minutes when Watson lobbed home after a through ball.
“I’m happy with the result,” said Braves manager Ricky Waddell. “I felt the sending off was very harsh. I felt Jack had two tackles – I understand why their player went down but I think he went down pretty early, in a sense that got Jack sent off.
"So I’m disappointed with that a wee bit. I can also understand from the ref’s point of view why he’s sent him off.
"I had a look at it again and I didn’t feel it was a yellow card. That’s with the benefit of hindsight. The guys dug in really well, I’m really proud of them.
"I thought we have defended over the last few weeks really, really well. We didn’t deserve to lose against Bonnyrigg but today we got our goals and we defended the lead.
"We played the vast majority of the game with 10 men so I can’t complain.”
Waddell praised his players’ composure in being able to comfortably seal the three points despite being a man down for so long.
"We’ve got that in our locker,” he said. “I think the guys have got a bit of quality.
"We needed a bit of calmness on the ball, we needed a bit of smart play at times.
"And I was getting awfully frustrated with some of the things we’re not doing at times.
"We just needed that calming influence. And we got that in fairness in the second half when we got the ball.
"They had a bit more of the ball in the second half, naturally because we were down to 10 men.
"But I’m so happy with the way the guys defended.
"We had a makeshift back four as well, with a right winger playing right back, a left winger playing left and our main centre back sent off.”
Mid-table Braves, who have 31 points from 23 games, are away to Celtic B in the league this Sunday, kick-off 3pm.