The Tykes scored all their goals before the break following an opening goal from Paddy McCourt and two from Chris O'Grady before the visitors made a game of it with two late goals from Albert Adomah.
After McCourt netted with a fine solo effort, Barnsley doubled their lead five minutes before the break when substitute Jim O'Brien's shot from the edge of the area was deflected past goalkeeper Jason Steele by O'Grady.
Eight minutes from the end, Middlesbrough managed to pull a goal back when Adomah fired in from 15 yards. Two minutes later he scored his second with a low header from four yards out but Barnsley held on.
Mowbray said: "It was 3-0 and we should have scored have scored some goals in that half. We had a couple of chances in the first half but we didn't take them.
"I've been showing them clips all week of the skill that McCourt has and they foul him.
"The second goal was a long-range shot that took a deflection. Things just don't go for us, and you wonder why we can't get one like that. The penalty was a rash decision from a young player.
"At half-time we needed to get some pride in the second half. We came out and took a couple of chances. I can't remember our 'keeper making a save. But (Lukas) Jutkiewicz hits the bar with only their 'keeper to beat.
"We have scored enough goals to be higher in the table but I have to solve the problem of goals going in."