Adam notched the decisive strike on 26 minutes - his second goal in three games following his winner against QPR - after Peter Crouch had headed down Ryan Shotton's cross.
Ryan Shawcross headed against the woodwork and Stoke had a late Kenwyne Jones' effort rightly disallowed for offside at the death as they deservedly claimed the three points to stretch their unbeaten league run at home to 13 games.
The win sees Tony Pulis' side climb up to 11th in the table on 16 points, the same total as Fulham, who stay ninth and are now winless in their last five games.
In driving rain, it was the visitors who made the brighter start, with Giorgos Karagounis lashing an effort across goal and wide on two minutes.
At the other end, Robert Huth headed wide from Matthew Etherington's corner on the quarter hour-mark and the Potters continued to pressurise, with Adam sending the ball over after juggling it in the area.
Moments later the Scotland midfielder scored, taking one touch in the six-yard box when Crouch headed Shotton's cross his way before swivelling and firing the ball in.
The hosts kept up the momentum as Glenn Whelan tried his luck from the edge of the area, drilling wide, and Shawcross almost made it 2-0 in the 36th minute when he clattered a header against the bar.
Fulham immediately broke forward, with Mladen Petric coming to meet a cross into the box but unable to finish, and Dimitar Berbatov - captain for the day in place of the suspended Brede Hangeland - then saw a delivery into the danger-zone scrambled away.
Soon after the restart, Whelan scooped an effort over from Jon Walters' tee-up while at the other end Karagounis had a shot balloon over.
Adam saw an effort deflect wide and Mark Schwarzer then tipped away a header by Huth from one Etherington corner, before Shawcross nodded another off target.
Fulham then went close as Berbatov's drive was saved by Begovic, while substitute Jones thought he had doubled his side's lead in the dying moments, but was flagged offside.