The impressive Adam turned and finished from close range in the 26th minute to register his second goal as a Potters player.
His first was the decisive strike in the previous home match, the 1-0 win over QPR on November 10, and Stoke are now unbeaten in 13 games on their own turf in the league stretching back to February.
Tony Pulis' side, who were the better team overall this afternoon and nearly had another goal when Ryan Shawcross headed against the bar, now have three victories to their name this season and are up to 11th in the table on 16 points.
Fulham, meanwhile, 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.
There was a shaky moment soon after for Stoke as goalkeeper Asmir Begovic misread a back-pass, but the ball was cleared away and the home side began to settle down.
Robert Huth headed the clearest opportunity yet 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 Peter Crouch headed Ryan 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.
Stoke looked eager to wrap things up as the second half got under way and seconds after the restart Whelan scooped an effort over from Jon Walters' tee-up.
Their opponents had ideas of their own though, and after Karagounis had ballooned over, Steve Sidwell dived to connect with an Ashkan Dejagah cross only for the ball to skim harmlessly off his head.
There was then a scare for Fulham goalkeeper Mark Schwarzer, who dropped a high ball but managed to smother it before anyone could pounce.
Adam, who was putting in a lively display, saw an effort deflect wide and Schwarzer then tipped away a header by Huth from one Etherington corner, before Shawcross nodded another off-target.
Fulham survived a scramble in the area after a Schwarzer punch, and then went close as Berbatov's drive was tipped away by Begovic.
Pulis had brought on Michael Kightly, who struck over, and with the visitors showing endeavour towards the end but unable to find an equaliser. Another Stoke substitute Kenwyne Jones thought he had doubled his side's lead in the dying moments, but was flagged offside.