With the score goalless, substitute Patrick Bamford was brought down for a 74th-minute penalty and stepped up to take the spot kick instead. Bamford, though, could only send his penalty over the bar and Stoke bagged the only goal of the game four minutes later on a night when second-placed Ipswich Town won 1-0 at Bristol City to leave Leeds nine points off the top two and having played a game more than the second-placed Tractor Boys.
Whites boss Farke made three big changes to the side that started Saturday's 3-2 comeback victory at Norwich City as summer signing Ilia Gruev, Jaidon Anthony and Willy Gnonto all came into the XI.
Bulgarian international midfielder Gruev was making his full Whites debut in replacing Glen Kamara who dropped to the bench along with Saturday's hero Crysencio Summerville and Dan James.
Leeds, though, made an extremely sluggish start and the Whites received the first of several first-half let-offs inside the first minute when Andre Vidigal could only fire straight at Illan Meslier after being picked out in the box by a cross from the right.
Meslier was then forced to save a flicked attempt from Ryan Mmaee with just five minutes on the clock and the Potters squandered a glorious chance on the quarter hour mark when a superb cross from Mehdi Leris picked out Vidigal who sent a free header straight at Meslier.
Two more saves from Meslier then kept out dangerous Stoke crosses before Mmaee lashed a wild shot over the bar but Leeds then created a golden opportunity themselves after brilliant play from Georginio Rutter.
The record signing turned brilliantly in midfield before supplying a beautiful through ball for Joel Piroe whose low shot was turned behind for a corner by keeper Mark Travers.
The Stoke custodian produced a good save to keep out a Rutter header from the corner but Stoke were soon back threatening themselves and the home faithful were left furious in the 31st minute after seeing claims for a penalty ignored.
Leris had turned Sam Byram in the area and ended up on the deck as the two players challenged but play was allowed to continue.
More danger then presented itself in the sole minute of first half stoppage time from a scramble as Archie Gray's attempts to usher the ball back to Meslier ended with Pascal Struijk clearing off the line.
Farke opted against making any changes during the interval and Leeds initially offered a very limited response but chances then began to present themselves.
Jaidon Anthony curled an effort wide from the edge of the box before Rutter fired a shot straight at Travers.
After intercepting an awful back pass, Rutter then dragged a shot from inside the box just wide before putting a header wide from an Anthony free kick.
Farke finally called for a triple change in the 70th minute as Summerville, Bamford and James were all introduced for Gnonto, Anthony and Piroe.
Summerville went close within seconds of coming on after being fed by Rutter before cutting inside and sending a curling attempt wide.
But Leeds were then presented with their best chance yet a few minutes later when Rutter played in Bamford who was brought to ground by Ben Pearson for a Whites penalty.
Bamford stepped up to take the spot kick but blazed his effort over the bar and five minutes later Stoke were ahead through a Pascal Struijk own goal.
StruijK had just cleared a dangerous free kick for a corner but Potters substitute Wesley headed against the bar from the corner and the ball then rebounded off Struijk at point blank range into the back of the net.
Leeds attempted to respond and Summerville looked sure to score after being played in by Bamford's flick only for Ki-Jane Hoever to save the day with last-ditch block.
Ian Poveda and Luke Ayling were then introduced from the bench and Poveda miscued a shot before Rutter again threatened but Leeds' chances had been and gone as Farke's side slipped to a third defeat of the Championship campaign.
Stoke City: Travers, Pearson, Rose, Burger, Vidigal, Johnson, Thompson, Hoever, Mmaee, McNally, Léris. Subs: Bonham, Gooch, Tchamadeu, Stevens, Laurent, Junho, Hakšabanović, Wesley, Lowe.
Leeds United: Meslier; Gray, Rodon, Struijk, Byram; Ampadu, Gruev; Anthony, Gnonto; Piroe, Rutter. Subs: Darlow, Ayling, Shackleton, Cooper, Kamara, Poveda, Summerville, James, Bamford.
Referee: Paul Tierney.