Despite being the Championship side, it was the Magpies who made most of the early running as they showed little sign of a hangover from Saturday's shock home defeat to Blackburn.
Matt Ritchie, Dwight Gayle and Mo Diame all went close and the Tigers struggled to cope with their attacking threat, as the sides eventually headed in level at the break.
The game continued to favour the visitors in the second half but they could not make the breakthrough and the most noteworthy moment of drama came from a late fracas, which saw Hull reduced to 10 men due to Dieumerci Mbokani's dismissal for violent conduct.
It meant extra time was required and Newcastle struck the first blow when Diame grabbed the opener after 109 minutes.
However, like London buses, after waiting all that time for a goal, the KCOM Stadium erupted just moments later when Robert Snodgrass netted an equaliser.
The game continued to ebb and flow, with Rafa Benitez's men repeatedly missing chance after chance but almost inevitably, the dreaded penalties were called upon to decide the winner as it finished 1-1 after 120 minutes.
Newcastle got off to the worst possible start as their first two takers Jonjo Shelvey and Dwight Gayle both failed to convert and with Hull scoring all three of their attempts, Yoan Gouffran's saw his effort saved by Eldin Jankupovic to book the Tigers' place in the semi-finals.