Riyad Mahrez opened the scoring with a well-struck free-kick on 23 minutes to send City ahead.
Chelsea continued to lack threat in the second half, and City capped a successful night late on as Mahrez netted from the penalty spot.
Pep Guardiola’s side now face the winner of Arsenal and League One side Oxford United, who play on Monday.
Sunday’s result means City beat Chelsea for the second time in a week — the first coming in a 1-0 victory in the Premier League on Thursday. City also knocked Chelsea out of this season’s Carabao Cup in November.
Information from Reuters contributed to this report.