The Clarets arrived at Oakwell as the only side in the Championship not having won on their travels this season.
They looked set for more misery when Adam Hammill fired Barnsley ahead in the 25th minute with his eighth goal of the season but the response from the visitors in the second half was impressive.
On-loan Manchester City striker John Guidetti nodded in a cross from Wade Elliott to draw the Clarets level in the 49th minute and Bikey headed in a corner five minutes later to secure his side's first win at Oakwell for 79 years.
Brian Laws has come under increased scrutiny in recent weeks and his side crossed the Pennines to face a in-form Barnsley side unbeaten in five matches, having won four of them.
Burnley, by contrast, had won only two of their previous eight games during a dire run that had seen them slip from fourth in the table to 12th in the space of two months.
Yet they almost led inside 14 minutes when on-loan Chelsea midfielder Jack Cork engineered the space to cross dangerously into the 18-yard box.
Sweden Under-21 international Guidetti, 18, arrived to connect with a firm header from 12 yards that hit a post as Barnsley survived.
Clarets winger Ross Wallace then tried his luck from 12 yards but saw his effort blocked by Tykes defender Jason Shackell.
Midway through the half, Barnsley began to enjoy a spell of pressure and they led in the 25th minute when Goran Lovre sent Hammill clear.
The highly-rated winger, whose performances have attracted admiring glances from a clutch of higher-placed clubs, saw his first shot pushed away by Lee Grant but followed up to sweep home the rebound left-footed from six yards out.
Burnley sought an equaliser before the break but Jay Rodriguez shot just wide from 12 yards out in the 35th minute.
Whatever Laws said to his players at half-time clearly had the desired effect as they were level within four minutes of the restart.
Elliott found space to cross dangerously from the right flank and Guidetti illustrated his quality with a fine header into the bottom left-hand corner.
Barnsley almost restored their lead three minutes later when Lovre headed against the post but Burnley survived and soon led themselves.
Wallace flighted in a corner from the right flank and Bikey arrived on cue to thunder home a header from 12 yards.
In the 64th minute, Rodriguez connected with a cross from Guidetti and headed against the post as Burnley sought a killer third goal.
Wallace then tried his luck from distance and Chris Eagles - a 72nd-minute replacement for Guidetti - went close with a 25-yard strike that drew a fine parried save from Luke Steele.