Callum Wilson's goal was his eighth of the season in all competitions
Callum Wilson's 16th-minute goal looked to be sending Newcastle into the last four until substitute Mykhailo Mudryk pounced on the veteran defender's error deep into stoppage time to send the game to spot-kicks.
The 33-year-old, who has suffered a dip in form recently, then fired Newcastle's second penalty wide and once keeper Djordje Petrovic saved Matt Ritchie's spot-kick Chelsea had gone from the despair of contemplating defeat to the sheer elation of a place in the last four.
It was cruel on Trippier, such an influential figure under manager Eddie Howe, but his fatal indecision on a header gave Chelsea that lifeline which Mudryk gratefully accepted.
Cole Palmer, Conor Gallagher, Christopher Nkunku and Mudryk were all on target for Chelsea but only Callum Wilson and Bruno Guimaraes scored penalties for Newcastle, who suffered fresh disappointment after their Champions League exit.
The draw for the semi-finals takes place after Liverpool v West Ham on Wednesday, with the first leg of those ties being played in the week commencing 8 January.
Following Mudryk's equaliser a male Chelsea fan climbed over the advertising hoardings before celebrating in the face of and bumping into Newcastle goalkeeper Martin Dubravka.
The fan was then led back into the crowd with referee Jarred Gillet speaking to stadium security before the shootout began.
-- Courtesy of BBC Sport