Kris Boyd grabbed the only goal of the game in the 52nd minute to earn a first league victory of the season for the Teessiders, managed by Speed's former Leeds team-mate Gordon Strachan.
Speed: "To come here and limit Boro to one clear chance, which is all they really had, is very encouraging.
"But the downside of that is that we didn't create too much ourselves. We had a few more chances than they did but didn't create a great deal which is something we'll have to work on."