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.
Although Stoke have now hit the 40-point mark long heralded by boss Tony Pulis as enough to guarantee safety, mathematically they cannot count their chickens just yet.
As for Norwich, six points clear of the bottom three, they too will still be looking over their shoulders, particularly with a vital game now looming large next Saturday against Aston Villa.
Chris Hughton's side will have to dramatically improve on this latest lacklustre away-day showing because with one victory on the road all season they are fortunate to find themselves out of the bottom three.
Not once did they test City goalkeeper Asmir Begovic, but then the moments of quality, as they were anyway, were few and far between.
It reached the point, at one stage in the first half, when referee Anthony Taylor was becoming the focal point of the game.
Taylor issued four bookings in the space of 15 minutes prior to the interval which came as a welcome relief.
There were chances, not that Begovic nor Mark Bunn for Norwich were ever put under pressure.
After only 40 seconds City captain Ryan Shawcross looped on to the roof of the net an Adam corner, whilst the latter drilled a 35-yard free-kick just wide of the angle of crossbar and post a few minutes later.
Norwich responded soon after with Robert Snodgrass planting a six-yard header over the bar in meeting a right-wing cross from Russell Martin.
Arguably the best moment of the game for Norwich followed shortly as Snodgrass, Grant Holt and Kei Kamara all combined to tee up Bradley Johnson for an edge-of-the-area left-foot shot that again cleared the woodwork.
After Peter Crouch back flicked a deep cross into the heart of the area from Adam on 15 minutes, Stoke's best chance of the opening half followed eight minutes later.
Again Adam was the supplier, this time with another 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.
After that the remainder of the half was littered with indiscretion, with Nzonzi the first into Taylor's notebook for a woeful challenge on Kamara.
Jerome followed for a high boot near the face of Bunn who lay prone for some time before suddenly springing to his feet, earning him the wrath of the Stoke fans with his every touch of the ball that followed.
Holt was also cautioned for clattering into the back of Robert Huth, as was Crouch for leaving a boot in on Kamara.
Mercifully, there was a reprieve on 42 minutes when the City crowd broke into a minute's applause for former Stoke midfielder Paul Ware who passed away 10 days ago - at the age of 42 - after a long battle against a brain tumour.
The half-time whistle could not come quick enough, but then at least 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 Huth found the head of 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 from seven yards.
With just three goals scored in their previous seven away games this year, there was precious little hope of a Norwich revival.
Instead, the Canaries 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.