Kenwyne Jones, making his first start for a month, gave the home side the lead after 12 minutes when he headed in from a corner.
It was 3-0 soon after when Ryan Shotton slotted in to an empty net from Jerome's square ball.
The half ended on a sour note for the Potters, however, when Jerome was given his marching orders after he had earned himself a second yellow card for an elbow on Yoav Ziv.
Ziv himself was sent off after the break when he kicked the air in frustration, only to see his boot fly off of his foot and hit the linesman.
Neither side posed any serious threat in the second half, but Stoke were happy to hang on for the victory.