Pressure is mounting on Davey after the Yorkshire side's miserable start has seen them lose five out of their opening six matches.
Davey and his crestfallen side left the pitch to the sound of boos and jeers from the 11,000 crowd following the dismal defeat.
A first-half goal from Peter Whittingham was enough to earn Cardiff victory, but Barnsley's best chances came in the dying minutes.
Cardiff , FA Cup semi-final winners over Barnsley at Wembley last season, breezed into a first-half lead, outclassing the Yorkshire outfit.
Barnsley, with four defeats in their opening five games, and languishing third from bottom, looked a pale shadow of last season's cup heroes.
The side who richly deserved their semi-final place after beating Liverpool at Anfield and Chelsea at Oakwell were absent again.
In stark contrast Cardiff, with an unbeaten start to the season, had the air of a top six side as they easily passed their way through the home side.
Home keeper Luke Steele kept them at bay when he saved Ross McCormack's 25-yard free-kick after 14 minutes but had no chance seconds later when Cardiff went ahead.
A cross from the left had the home defence in a tangle and the ball broke for Whittingham, who stroked in a cool finish from 12 yards.
Cardiff could have stretched their lead before the break when a slick move sent in Gavin Rae, but Steele dived full stretch to turn away his low shot.
And when ex-Barnsley midfielder Stephen McPhail struck a pinpoint free-kick from the left Roger Johnson's towering header beat Steele but hit the bar.
Cardiff endured a nervy finish as they were forced back when Barnsley hunted for the leveller and produced their best chances.
Last season's 14-goal top scorer midfielder Brian Howard was just wide with a clever 77th minute shot.
And eight minutes from the end Kaode Odejayi, scorer of last season's FA Cup winner over Chelsea, went close with superb header from Stephen Foster's cross.
External Reports
Wales Online