Gylfi Sigurdsson put the hosts ahead with an early contender for goal of the season, but the game went into extra-time following a horrible gaffe from Brad Friedel.
Paul McShane put Hull on course for victory in extra-time, but Harry Kane made it 2-2 before Friedel saved Ahmed Elmohamady's penalty in a marathon shoot-out to set Spurs up with a quarter-final clash against West Ham.
Newcastle deservedly took Manuel Pellegrini's side to added time and probably had the better of things over 90 minutes, but they could not break the deadlock and instead left St James' Park digesting an unflattering 2-0 defeat.
And while City's strike pair must take their share of the praise for neat finishes in the 99th and 115th minutes, Costel Pantilimon also did everything that was asked of him having been brought in to replace Joe Hart.
Capital One Cup quarter-final draw
Leicester v Manchester City
Stoke v Manchester United
Sunderland or Southampton v Chelsea
Tottenham v West Ham
- Ties to be played week commencing December 16.