A double from Sam Clucas gave the away side at Oakwell three huge points to lift them off the foot of the Sky Bet Championship table, while Barnsley dropped to the bottom with only one win this term.
The opening goal came within just seven minutes as Brad Collins in the Barnsley goal passed the ball straight to Clucas, who chipped the ball over the keeper from over 35 yards out for his fourth goal this season.
Amid some boos from the home supporters, James McClean was threaded through on goal, but his left-footed shot forced Collins into a save.
The first real chance for the hosts was a long-range effort from Kenny Dougall, but his shot veered wide.
Stoke then doubled their advantage from the penalty spot. After some neat play down the left the ball fell to McClean, who was brought down by Alex Mowatt, with Lee Gregory converting by slotting the ball into the bottom left corner.
The Tykes had the chance to draw level when Conor Chaplin went to ground in the area and the ball fell to Mowatt, whose shot glanced wide.
Stoke came close to increasing their lead when Gregory's shot from six yards out bounced off Collins.
The resulting corner gave Stoke their 3-1 lead. The ball bounced around the area and fell to Allen, who guided the ball into the bottom left corner.
Three minutes later, Clucas got his second goal of the game, shooting from distance once again to give Collins no chance.
Barnsley pulled a goal back, after a high through ball found substitute Patrick Schmidt, who placed the ball into the bottom corner.