First-half goals from Lyle Taylor and Andre Green had seemingly set Lee Bowyer’s side on course for a comfortable home victory.
But Cauley Woodrow pulled a goal back after 72 minutes, sparking belief that the visitors could find a leveller.
Struber, however, said: “It is always the same mistakes, the same as last week at Portsmouth. Set plays were the difference again and that is why we have lost.
“The problem is that set plays are now a very important part of football and you have to be able to defend them.
“In every other phase of our play, how we build up the play and the way we create chances, we are ready for the Championship, but it is clear that we are not ready for the Championship from set plays. This is the reality. We don’t stay in our structure at them.
“We spent a lot of time on set pieces during the week but at the end we made no progress in them.
“It was a big game for us, a six-point game. That was very clear to us before the game.
“We hit the post and crossbar three times from chances that we created, so that makes it even more disappointing.
“When I see what we can do, what we can create, how we had good pressure on Charlton in the second half, got good rotation of the ball and hit the crossbar three times, that makes me all the more disappointed.
“This defeat was not necessary and we were a bit unlucky in the end, but I have no words to describe my feelings now.
“We brought good players into the club in the transfer window. We have to work hard now – and I believe that we can stay in the league.”
Charlton manager Bowyer acknowledged the importance of squeezing past Barnsley as his side stayed clear of the relegation pack at The Valley.
With a trip to Stoke to come neon Saturday and games against fellow strugglers Luton and Huddersfield to follow in a packed February, Bowyer was pleased his side maintained their four-point cushion over the bottom three with a 2-1 home win.
Charlton Athletic: Phillips, Matthews, Lockyer, Pearce, Sarr, Doughty (Purrington 90), Cullen, Williams (Forster-Caskey 57), Pratley, Green, Taylor (Hemed 68). Unused substitutes: Amos, McGeady, Field, Oztumer.
Barnsley: Radlinger, Jo Williams (Simoes 68), Sollbauer, Andersen, Oduor (Ludewig 68), Thomas, Halme, Mowatt, Brown, Woodrow, Chaplin (Schmidt 86). Unused substitutes: B Williams, Dougall, Walton, Styles.
Referee: J Brooks (Leicestershire).