The goal was one of real quality. It was scored by Junior Mondal, who gave goalkeeper Josef Bursik little chance with a powerful drive angled into the top corner of the net.
However, far from setting Boro up to go for more, the visitors were pushed back for long spells by a young Wimbledon side who made life very difficult and almost drew level when Tony Sibbick headed inches over, rounding off a well-worked routine from a corner.
Joao Morelli was denied a goal by a smart save from Bursik, but it was the home side who looked more dangerous and after Joe Fryer had saved well from Egli Kaja, the Dons went close three times with shots that flashed just off target.
Boro made life difficult for themselves when reduced to ten men as central defender Matthew Elsdon was sent-off for two bookable challenges in a very short space of time. After losing their edge and having to reshape, they were glad for half-time when they were able to calm down, take stock and reorganise.
Chances were few and far between in a second half that was niggly and patchy. Paul Kalamabayi headed just over for the Dons and Ben Liddle saw a shot blocked as Boro looked lively on the break, but it was defensive work that was called for as Boro saw out the game despite being a man down.
They needed to dig in and work, they did just that to take their second win in the competition following on from a 1-0 win over Charlton which came after an opening game defeat at Fulham.
Boro are part of a four-team group where each team plays each other twice, and the top two progress to the knock-out stages.
Boro: Fryer; Hegarty, McGhee, Elsdon, Kitching, McGinley (Morris), Wheately, Cooke (Capt.), Liddle, Morelli (Maloney), Mondal. Subs: Pears (gk); Coulson, McAloon.