Adam's first goal since November 24 was enough to further ease the Potters' relegation fears following last weekend's 2-0 win at QPR.
Norwich, six points clear of the bottom three, will still be looking over their shoulders, particularly with a vital game now looming large next Saturday against Aston Villa.
Stoke's best chance of the opening half came after 23 minutes with Adam supplying a corner that was struck on the volley by Cameron Jerome from eight yards out and into a crowd of players in front of him.
There were muted cries of handball but it was not a Norwich arm that was in the way, instead that of Stoke midfielder Steven Nzonzi.
The second period was spiced up after just 50 seconds.
It was typical route one stuff from Stoke as a long ball pumped forward from Robert Huth found the head of Peter Crouch for a knockdown into the area. As Jerome blocked off Sebastien Bassong's attempt to clear, Adam stole in for a simple left-foot shot underneath Bunn.
Norwich should have been dead and buried in the 63rd minute but for a miss-of-the-season from Crouch who failed to connect from four yards out with a low ball into the six-yard box from Adam.
Comically, six minutes from time Crouch missed another sitter from eight yards after Bunn had blocked a stinging drive from Nzonzi.
The gangly striker could afford a smile but only just as in injury time Kamara planted a header inches over from a long throw in what was City's only meaningful second-half attempt as Stoke deserved the win.