Signed from Brentford on a one-month loan deal just prior to the trip to South London, 22-year-old winger Dallas scored the second goal after Lee Collins had earlier netted Town's first on the road this season.
Although the Cobblers remain in the bottom two in the League Two table the win is a huge boost to morale after a week in chairman David Cardoza had expressed criticism of the squad while backing manager Aidy Boothroyd.
His opposite number, Neal Ardley, was far from happy with AFC Wimbledon's display, telling BBC London: "I challenged them prior to the game, and said it's a game we should win and it would tell us a bit about what we have to do and what aspirations we should have this season.
"We knew what they would do and my staff and I prepped them on everything and I can't defend my players. It was unacceptable."
Dallas, meanwhile, said: "It means a lot to get my first goal. Every attacker wants to score goals but the main thing is to get the three points."