After an impressive start to the game, the home side took an early lead on 11 minutes through Skuse. Some good work from Stead and Wood created the chance for the midfielder.
The game eased off following the goal, with Bristol holding that important lead.
Barnsley slowly came back into the game and had two attacks wrongly ruled out for offside, much to the frustration of the away fans.
The sides went in at half time with the score at 1-0.
The second half began with Bristol City looking the stronger of the two sides.
On 51 minutes, the home side had the chance to double their lead after winning a penalty. Stead smashed the spot kick into the top left to make it 2-0.
A great run from Adomah set up Stead, who then forced Button into another save.
Bristol City saw out the remainder of the game and eased to a 2-0 victory, earning all three points.