Alfred N'Diaye and Conor Coady were denied by the Chile goalkeeper, while the hosts scored all four of their attempts in the shoot-out.
Sergio Aguero was chasing City's goalscoring record, having equalled it against Burnley on Saturday by moving on to 177 goals, but endured a frustrating evening until scoring the winning penalty with a cheeky Panenka.
The drama came after neither team could find a breakthrough as a tense, end-to-end battle over 120 minutes ended 0-0.
