Pogrebnyak joined Martin Jol's side from Stuttgart on transfer deadline day and quickly made his mark at Craven Cottage.
Despite training with his new team-mates for the first time two days ago, the Russia international fitted in seamlessly and, at first glance, seemed a more than ample replacement for Bobby Zamora as he netted 16 minutes into his debut.
The 28-year old controlled a wayward John Arne Riise shot to rifle home, before Fulham double their advantage as Clint Dempsey's superb 30-yard drive rebounded off the bar onto goalkeeper Thomas Sorensen's back and in.
Pogrebnyak's afternoon ended in the 62nd minute as he hobbled off injured and Stoke ensured a nervy ending to the match when captain Ryan Shawcross reduced the deficit from a corner.
However, Fulham held on for a fifth win in seven home games as Stoke fell to a fourth successive league defeat.
Both sides made four changes apiece for the match, which began slowly on a bright afternoon by the Thames.
Neither team proved a real threat during the opening exchanges, although Mousa Dembele was impressing for the hosts despite playing in an unusual role alongside Danny Murphy in central midfield.
The Belgium international was leading the City defenders a merry dance and put an incisive ball behind the Stoke backline but Damien Duff's cross was stopped by Sorensen.
Fulham were soon in the ascendancy and, aside from a tame long-range Glenn Whelan effort that Mark Schwarzer held with ease, Jol's side were comfortable.
The Cottagers turned their dominance into a 16th-minute goal as Pogrebnyak controlled a wayward Riise inside the box and unleashed a fierce drive into the top left-hand corner.
He was fortunate not follow up his debut goal with a booking shortly afterwards for a late challenge on Wilson Palacios, who soon dusted himself down to shoot just wide.
Kenwyne Jones was another pushing for a Stoke equaliser, although that task was made harder in the 28th minute.
Out of the blue, in-form Dempsey fired a 30-yard effort that rattled the crossbar and rebounded onto goalkeeper Sorensen - and in off his back.
The second goal left Stoke looking devoid of ideas going forward and they had Shawcross to thank for stopping a third at the other end.
Pogrebnyak was proving difficult to handle and darted down the left, before firing a cross into the six-yard box that would have been tucked home by Dempsey was it not for the Stoke skipper's clearance.
Dempsey became the game's focal point as half-time approached, with Marc Wilson and Palacios picking up bookings for challenges on the American midfielder.
Murphy curled just wide from the first free-kick on the edge of the box, before the Fulham skipper's chipped ball into the box would have been tucked home by Duff had Sorensen not been alert to save.
Stoke looked rejuvenated immediately after the restart and came close to reducing the deficit through Matthew Upson, who volleyed Whelan's free-kick just wide in the 47th minute.
The visitors were by no means dominating second-half proceedings, though, and a Stephen Kelly cross almost crept inside Sorensen's near post moments later.
The match was began to ebb and flow more as the half wore on, before Fulham were forced to change things.
Pogrebnyak appeared to fall awkwardly in a challenge with Shawcross and had to be replaced in the 62nd minute by Simon Davies, which resulted in Dempsey moving into attack.
Stoke made changes of their own in a bid to change the course of the game shortly afterwards as Pulis used his full complement of substitutes.
Ricardo Fuller, Jermaine Pennant and Ryan Shotton all came on and the changes soon had the desired effect.
Aaron Hughes deflected a Fuller shot out for a corner, which was crossed in by Pennant and flicked home at the near post by Shawcross.
Peter Crouch flashed a shot over from close range as Stoke pushed for a leveller, leading Jol to bring Dickson Etuhu on for Murphy in a bid to shore things up.
Sorensen had to react quickly to prevent Dempsey netting a last-gasp third after a defensive mix-up, before the United States international hit the post after rounding the Danish goalkeeper at the end of the five minutes of stoppage time.