Barnsley have not won in nine games after they lost lost 1-0 at Bradford City on Saturday.
An early goal was enough to defeat the Reds who failed to muster a shot on target until injiury-time.
Barnsley slipped down to 13th in the League One ahead of a final day game next Sunday against Rochdale who can no longer get into the play-off positions.
It was an emotional afternoon as fans of both clubs remembered the Valley Parade fire in which 56 people died 30 years ago this week. England manager Roy Hodgson attended the match in which Barnsley wore a special white kit and laid a wreath of 56 white roses.
Reds head coach Lee Johnson made four changes with the injured Sam Winnall and George Smith dropping out of the squad as did Jack Cowgill and Milan Lalkovic. They were replaced by Jabo Ibehre, Declan John, Ben Pearson and captain Martin Cranie.
Jon Stead tested Adam Davies with an early shot then put the Bantams ahead on 16 minutes when he brilliantly volleyed in from ten yards out after meeting a cross from the left. It could have been two goals in two minutes when another cross from the left exposed the visiting defence and and Christopher Routis fired wide when well-placed ten yards out.
All the Reds offered before the interval were three long-range shots which Conor Hourihane blazed high and wide as well as an off-target header from Ibehre.
Luke Berry and Paul Digby fired well wide from long-range early in the second half then Routis had another glorious chance when he misjudged a right-wing cross and headed over the bar inside the six-yard box.
Bradford goalkeeper Ben Williams dropped an impressive Hourihane free-kick late on but Mason Holgate could not convert from a tight angle.
Teenager George Maris made his Barnsley debut late on and had their only shot on target with a looping headed into the goalkeeper's arms on 93 minutes.
