Owls substitute Atdhe Nuhiu scored the only goal of the game with an 87th-minute penalty after Boro midfielder Dean Whitehead was sent off for handling Leon Best's goal-bound shot inside the six-yard area.
Linesman Paul Hodskinson, who flagged to award the late spot-kick, earlier failed to spot that Jacob Butterfield's drive from distance crossed the line after hitting both posts.
The result puts Wednesday one point and one place behind their mid-table opponents, who failed to score for the seventh successive game.
Karanka said of Butterfield's 'goal': "I can't say anything because I couldn't see the ball. For me, the linesman couldn't see the goal in our favour and he could see the penalty. This is the problem.
"We lost the game because we didn't play well. I don't have to put excuses on the referee or linesman. We had four or five chances to score; we had the post chance and Danny Graham and Kei Kamara had very, very clear chances and if we had scored a goal we would have won the game."
Middlesbrough last scored in mid-January and Karanka said his side's lack of goals was a worry.
"It's a big concern for me," he added. "This team, with the same players, scored a lot of goals in the past. We won five or six games in a row and had a lot of draws.
"We won against Millwall two-nil and against Blackpool. We scored two goals at Bolton, won three-nil at Reading and for this I am concerned because this team is capable of scoring goals and we're seven games without scoring."
Sheffield Wednesday head coach Stuart Gray was pleased to secure all three points from an 'ugly' game.
"I think we deserved the win," he said. "The disappointing thing was the first half - there was no intensity.
"One thing you do know is that Middlesbrough are going to have a lot of possession of the ball so we tried nullifying that by pressing them up high but then it was one, two, maybe three touches too much instead of moving the ball quicker.
"If you'd said to me before the game: 'Play ugly and win one-nil', I'd have snatched your hand off."
Gray also called for goal-line technology in all leagues following the Butterfield shot and a similar incident which went against his side in Monday's FA Cup defeat to Charlton.
"I haven't seen the incident, I just felt that linesman was in the correct position," he said.
"It just shows that you can put everything to bed if everybody's got goal-line technology. I looked at their players' reactions and thought it must be close. Their dugout were up and they thought it was in.
"It's crazy how we're talking about this situation again (after the Charlton game).
Maybe that was that bit of luck we deserved; it balanced itself out because we didn't get that luck on Monday night.
"As for the penalty, I think the boy has stopped it with his hand.
"I think Andy D'Urso is probably a little bit blocked because we had that many bodies in their box and the linesman's had to help him out with the decision. I looked straight away and he put his flag across his chest."