The result, which was 1-1 at full-time before spot-kicks, means the Blues are now in the semi-finals of the competition.
Newcastle took an early lead through Callum Wilson after some calamitous Chelsea defending, but it was later Trippier's turn to level things up.
The England right back botched a headed clearance in the second minute of stoppage time, with Mykhailo Mudryk pouncing on the error to poke home the equaliser.
Then, in the shoot-out, Trippier missed his effort, before Dorde Petrovic saved from Matt Ritchie to end the contest.
The experienced right-back is unlikely to face much criticism, being forced to play on amid an injury crisis for Eddie Howe's side.
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However, recent weeks have been error-strewn, and one of the tougher periods of the former Tottenham and Atletico Madrid star's career.
Joining Chelsea in the final four are Middlesbrough, who beat Port Vale 3-0, and Fulham, who overcame Everton on penalties.
The final spot in the semi-finals will be decided tomorrow when West Ham travel to Anfield to face Liverpool.