A 73rd-minute strike from Conor Chaplin extended Boro's alarming winless streak to eight matches as they produced another feeble performance following the previous weekend's inept showing against lowly Luton and another poor showing against another struggling side in Wigan Athletic.
It was an Oakwell afternoon when inept Boro failed to muster a single effort on target.
Damning chants of 'You're not fit to wear the shirt' were directed towards Boro's players from the furious 3,999 travelling contingent as they headed off the Oakwell pitch .
Angry scenes followed the final whistle with substitute Rudy Gestede appearing to gesticulate towards supporters as the away players headed past the away end into the changing rooms in the North Stand.
Woodgate, whose side are now just three points above the relegation zone and have not won in the league since New Year's Day, said: "It was an absolutely fantastic support and I don't blame that for being angry and going crazy. I expect that if we don't win games like this.
"We are going to get hard times as the fans are just as frustrated as we are. I appreciate all the support that they give the team and they are going to express their opinions to us and we must take it on the chin.
"I have players in that dressing room who I am sure will turn it around. I have no doubt about that. You may think I am crazy thinking that after defeats against Barnsley and Luton. But I truly believe that. We will turn it around.
"We went through that spell where we were playing really well and got ourselves ten points clear. Now we are three off it and need to start winning games quick.
"They (Barnsley) played well and we weren't great and did not create any opportunities and we did not have that believe to credit chances.
"We had a big following and did not give them anything."