Dimitar Berbatov's second-half winner gave the Reds three vital points in the title race, but the succession of chances passed up en route to victory left the United manager befuddled after the match.
"We’re concerned that we are making so many chances and not taking them," Sir Alex told MUTV. "In the first half, I don’t know how many we had.
"I hope we don’t live to suffer because of it. If there’s one part of the game that’s giving us any concern, it’s the finishing. Their performance level is very good, and the concentration and energy of the team are very good as well, so we’ve got to hope we can start scoring regularly."
Fortunately for Sir Alex and his players, the Reds' miserly defence ensured that a solitary goal was enough for victory. United have now gone seven Premier League matches without conceding a goal, since November's 2-1 defeat at Arsenal.
Boro were kept out despite the absence of Rio Ferdinand and Patrice Evra, and Sir Alex was keen to single out rookie central defender Jonny Evans for special praise.
"We are very good defensively," he said. "Jonny Evans has come in for Rio Ferdinand, and he’s done fantastic. The boy’s an absolute gem of a player. We’re so lucky to have him. He’s only 20, but he reads the game so well."