The hosts, mired in the relegation zone, dominated the contest but failed to make their chances count against a Rovers side coming off the back of three straight league defeats.
Barnsley's best chance to make the breakthrough came inside the first 20 minutes when they were awarded a penalty after Bongani Khumalo brought down Marcus Pedersen, but Chris O'Grady saw his spot-kick saved by Turnbull.
The result made it three draws in a row for Barnsley, stretching their unbeaten run to four matches.
The stalemate ends Rovers' run of three defeats - but they nearly snatched all three points in the last minute, Ritchie Wellens curling a shot from the edge of the area just wide of the far post.
Before the match Rovers announced that John Ryan had resigned as chairman after 15 years at the club - which began with Rovers in the Conference.
He watched the first half from the directors box and sat among the Rovers' fans after the interval. Throughout the match the travelling support sung his name.
Barnsley had the first chance of the match in the 11th minute when an unmarked Jean-Yves M'Voto headed a Paddy McCourt corner wide from six yards.
After Turnbull's spot-kick save there were few chances until eight minutes before half-time when Mark Duffy's cross nearly embarrassed Barnsley keeper Jack Butland as it dipped at the last moment.
Two minutes later McCourt's solo run took him past four defenders and into the Rovers area but he hit his shot straight at Turnbull.
In the 53rd minute Rovers striker Chris Brown was brought down by Barnsley captain Martin Cranie as he broke into the Barnsley half and was booked.
A low shot from Pedersen from the edge of the area forced Turnbull to make a low save. Two minutes later Pedersen was unable to get a touch on a cross from the right by Tom Kennedy.
Barnsley thought they had won their second penalty of the match when McCourt went down in the Rovers' area after a run but referee Kevin Friend gave a free-kick to the home side.
Fifteen minutes after Cranie was booked, Rovers' Dean Furman was booked for a foul on David Perkins.
Rovers' full-back Paul Quinn had to turn a Perkins cross from the right behind for a corner with Pedersen breathing down his neck.
A left foot shot from the edge of the area by O'Grady was well caught by Turnbull.
Eight minutes from the end Rovers' Reece Wabara turned a Jim O'Brien cross into the side-netting, before the visitors' striker Brown became the third player booked when he kicked the ball away after the referee had blown for a foul.