After a fascinating 90 minutes which saw the tie end 2-2 on aggregate, the tie was settled when David Luiz scored the winning kick after Kepa Arrizabalaga’s fine save from Lucas Moura.
Both sides had chances to win it before spot-kicks, but in the end, it came down to a test of nerves from 12 yards with no extra-time or away goals in the semi-finals this year.
Chelsea, who have now reached 15 finals in 16 years in all competitions, will face Pep Guardiola’s City at Wembley on 24 February.
Willian, Cesar Azpilicueta, Jorginho, and David Luiz all kept their cool, while Dier blazed over and Kepa kept out Moura with a full-stretch dive to his right, sparking delirium inside the Bridge.
Blues boss Maurizio Sarri, who said after losing to Arsenal at the weekend that his players are “extremely difficult to motivate”, celebrated with his team at the end.