The narrow victory moves Tony Pulis' side onto 40 points for the season, eight clear of the relegation zone, thanks to Adam's crucial goal at the start of the second half.
Norwich failed to produce a shot on target in the entire game and their best opportunity came inside the opening 10 minutes when Robert Snodgrass headed Russell Martin's cross over from six-yards, before Bradley Johnson pulled a low shot wide.
Cameron Jerome almost put Stoke ahead after 22 minutes when he met Adam's corner at the far post, only to see his shot blocked by teammate Steven N'Zonzi on its way in.
N'Zonzi and Peter Crouch were both lucky to only receive bookings for challenges on Kei Kamara and Alexander Tettey respectively after a scrappy end to the first half.
One minute after the restart Crouch's cushioned header fell into the path of Adam who raced into the box and drove a low shot past Mark Bunn, his first goal for the club since November.
Crouch should have doubled Stoke's advantage after the hour mark when he hit a fresh air shot from three yards out after more good work from Adam, while a series of attacking substitutions from Chris Hughton failed to inspire Norwich at the opposite end.
Kamara saw a late header loop onto the roof of the net and that was the closest the Canaries came to equalising as they remain on 38 points for the season in 14th position.