Jason Shackell and Adam Hammill were on the scoresheet on a miserable afternoon for big-spending Boro, whose supporters were quick to express their displeasure at the end of the game.
Strachan admitted: "We're disappointed to make so many chances and not get anything out of the game.
"We had enough chances to get something out of the match. We had the better start but they scored at the right time.
"They were dogged and determined and played well. Their keeper (Luke Steele) played well and he must be pleased with himself. We had four passing players missing but the ones who played gave it a real go.
"I can't complain about that but we'll have to better than that. The ability to pass the ball is a priceless asset in football and we're not doing that well.
"I don't have any problems with the fans jeering us at the end."
Barnsley chief Mark Robins felt the Tykes deserved all three points and praised the performance of Steele. "I am absolute delighted with the performance. I thought we were excellent from start to finish.
"We had a spell in the second half where we got stretched. Everybody played a great part in a great display.
"I thought we were good value for the win. We have to compete on the pitch when we play sides like Middlesbrough, it is always difficult for clubs like ourselves to compete in the transfer market.
"We defended well where it mattered. Last week we were garbage and today we nearly got up to caviar. Luke Steele made some good saves after last week's criticism.
"We played some decent stuff and we attacked with more of a purpose. It showed what good players we have. The next set of games are big ones for us. We'll enjoy this while it lasts."
Barnsley: Steele, Hassell, Foster, Shackell (O'Brien 59), McEveley, Trippier (Butterfield 90), Colace, Arismendi (Doyle 79), Hammill, Lovre, Gray. Unused substitutes: Preece, Hume, Morales Neumann, Devaney.
Middlesbrough: Steele, Bates, Wheater, McManus, Kilgallon (Luke Williams 81), Robson, Bailey, Arca, Kink (Halliday 66), McDonald, Boyd. Unused substitutes: Coyne, Lita, Miller, Hoyte, Smallwood.
Referee: K Hill (Hertfordshire).
Man of the match: Luke Steele