Hull had lost 1-0 to Spurs in the Premier League on Sunday and they came within a whisker of exacting swift revenge on their return to White Hart Lane.
Icelandic midfielder Gylfi Sigurdsson was one of the players brought into the Spurs team and he made a stunning impact in the 16th minute, flummoxing his marker with a neat drag-back and then arrowing a sumptuous shot home from 25 yards.
YouTube: Pedroka Henri
Second-choice goalkeeper Brad Friedel made a less favourable impression, however, gifting Hull an equaliser in the 53rd minute.
Liam Rosenior’s low cross from the right was hooked back across goal by the sliding Curtis Davies and Friedel inadvertently fumbled the ball into his own net for an own goal.
Spurs saw 20-year-old midfielder Harry Kane rattle the woodwork from 20 yards in stoppage time, but it was Hull who struck first in extra time, with Paul McShane heading home from a George Boyd corner.
Kane would not be discouraged, however, and he drilled home a 108th-minute equaliser to make it 2-2 and send the game to penalties.
Aaron McLean missed for Hull before Spurs’ record signing Erik Lamela fluffed his lines for the hosts, but Friedel had the final say, saving with his legs from Ahmed Elmohamady to secure a nail-biting win.
Spurs’ reward was a home time with West Ham United, while Manchester City will travel to second-tier Leicester City after Alvaro Negredo and Edin Dzeko struck in extra time in a 2-0 win at Newcastle.