Dempsey netted his eighth and ninth goals of the season as the Cottagers made it three wins in five to put daylight between themselves and the drop zone.
The game was the third tempestuous affair between the sides this season and Potters captain Ryan Shawcross was sent off after bundling over Dempsey in the box.
The American, who had already given his side a half-time lead, stepped up to convert the resulting penalty and seal a comfortable victory.
The opposing managers had refused to shake hands following their two previous meetings this season but Mark Hughes was happy to greet Tony Pulis after this one.
The home fans were not so generous, repeatedly jeering Andy Wilkinson for the September tackle on Moussa Dembele that initally sparked the bad blood between the sides.
Pulses were racing from the off, with Shawcross nodding Matthew Etherington's free-kick inches wide inside five minutes and Damien Duff crashing a terrific 30-yard strike against the crossbar two minutes later.
Danny Murphy was narrowly off target after being invited to shoot from 20 yards before David Stockdale saved Tuncay's low strike from the tightest of angles.
Both sides then began to toil until Duff tried a 22nd-minute curling shot which an unsighted Asmir Begovic palmed behind, before Murphy was booked for a tackle from behind on Dean Whitehead.
Stoke almost scored from the resulting free-kick, with Chris Baird's mistimed clearance falling to Kenwyne Jones, who blasted the ball against the near post.
The febrile home fans were incensed by a number of decisions that went against their side but they were celebrating in the 33rd minute when Dempsey gave them the lead.
Duff pounced on a loose ball as Stoke went to sleep, allowing Andy Johnson to get to the byline and cross for Dempsey to slide home his eighth goal of the season ahead of Wilkinson.
A frustrated Wilkinson was booked two minutes later after needlessly clattering into Johnson on the byline.
Fulham enjoyed the better of the early moments in the second half without really threatening.
Referee Stuart Attwell brandished a red card and Dempsey blasted the penalty straight down the middle.
The 10 men quickly saw a double change, with new signing John Carew and Abdoulaye Faye on for Tuncay and Danny Collins.
Fulham then lost Dickson Etuhu to injury, meaning a debut for Steve Sidwell, with Stoke responding by withdrawing Etherington for Glenn Whelan.
Wilkinson's speculative strike deflected behind off Rory Delap, while Sidwell's more measured effort also took a nick at the other end.
Johnson should have made it 3-0 18 minutes from time but shot straight at Begovic under pressure from Faye, before Delap nearly deflected a mis-hit clearance into his own net.
Zoltan Gera and Rafik Halliche then replaced Dembele - who had been impressive on his first start for more than two months - and Aaron Hughes for the final 15 minutes.
Jones nodded Delap's corner over the top as the visitors made a token effort to set up a tense finish, but they were long since beaten.
