The visitors had the first real chance of the match when Russell Martin's cross found Robert Snodgrass six yards out, but the former Leeds United man could only head his effort over the bar.
The hosts almost took the lead in the 23rd minute when Cameron Jerome's strike from close range looked destined for the net, only for Steven N'Zonzi to unwittingly block his teammate's effort on the line.
Stoke did break the deadlock right at the start of the second half when Peter Crouch nodded the ball down to Adam, who fired his shot underneath Mark Bunn.
Crouch could have got his own name on the scoresheet just after the hour mark when he had the goal at his mercy, but he failed to make any contact with Adam's cross.
It was almost 2-0 with five minutes left to play when N'Zonzi found space to shoot, but Bunn was on hand to make a fine stop.