Ross McCormack's goal five minutes into the second half sent Leeds into the last 16 and ensured Spurs befell the same fate as Norwich City and Queens Park Rangers, who were knocked out on Saturday.
Luke Varney had given the Whites a first-half lead and, though Clint Dempsey pulled a goal back, Spurs were unable to find an equaliser as some brilliant last-ditch defending denied Jonathan Obika in the last minute.
Leeds were in front through the game's first real chance on 15 minutes as Michael Brown's pass sent Varney clear of the Spurs defence and he kept his composure to finish past the recalled Brad Friedel.
Spurs' response saw Gylfi Sigurdsson head over and Dempsey flicked a shot straight at Jamie Ashdown, while in between, Varney fired over for Leeds.
More slack defending from Spurs presented the hosts with another opportunity just before half-time, but Friedel was out smartly to deny McCormack.
However, there was nothing the veteran keeper could do five minutes after the interval when McCormack raced on to El Hadji Diouf's through ball before arrowing a left-footed shot past Friedel after cutting inside Steven Caulker.
The Premier League side were much more purposeful in their response to conceding the second goal as former Leeds winger Aaron Lennon's dangerous cross was turned wide by Dempsey.
Dempsey was more clinical on 58 minutes, though, darting to the near post to glance Gareth Bale's left-wing delivery beyond the reach of Ashdown.
Andre Villas-Boas promptly introduced Obika and Mousa Dembele, who saw a shot from outside the box deflected wide on 71 minutes.
But, just 60 seconds later, Spurs could have fallen further behind after McCormack scampered clear of the defence again, only to see his shot strike Friedel's leg.
Spurs had one last chance to force a replay after Benoit Assou-Ekotto's raking long pass sent Obika clear, but the substitute took too long to bring the ball under control and Lee Peltier produced a superb block to ensure Leeds' progression.