Adam latched onto a Peter Crouch knockdown and swept the ball past Mark Bunn to secure a vital victory in a scrappy game at the Britannia Stadium.
The Potters have now reached the 40-point mark that has always been Tony Pulis' target, while the Canaries could still need one more win as they remain down on 38 points.
Robert Snodgrass had the first opportunity of the game when he headed over the bar despite being well placed to put Norwich into the lead.
Bradley Johnson then struck a shot over the bar but it was Stoke doing most of the pressing as they lofted a series of corners and free-kicks into the box.
Cameron Jerome went closest to finding the breakthrough when he hooked an Adam corner towards goal but the ball struck the arm of team-mate Steven N'Zonzi.
N'Zonzi did not have the best of halves and shortly afterwards he was a little fortunate to get away with just a booking after needlessly scything down Kei Kamara.
That prompted a scrappy passage of play in which Jerome, Grant Holt and Crouch were all booked for poor challenges as the first half ended on a sour note.
But the home fans soon had something to cheer after the interval. Crouch headed on a long ball from the back and Adam nodded it on with his head before half-volleying neatly to score.
Adam attempted to return the favour soon after as he won the ball on the right before squaring it to Crouch but the striker inexplicably missed the cross completely when unmarked in the middle.
After that flurry of excitement the game reverted to type with a series of fouls adding to the stop-start nature of the contest.
Norwich offered little in attack with Asmir Begovic catching numerous crosses and Stoke held on with relative comfort to surely ensure another season of Premier League football for the Potters.