Billy Daniels put City ahead in the 29th minute with a lovely finish before Josh Ruffles added a second three minutes later.
New boy Courtney Richards played in the match, dominated by a Sky Blues team who excelled in every position, and he scored from twelve yards in the 50th minute with a reactive strike.
The fourth came through Will Roberts ten minutes later and late on, Jonson Harris had a goal ruled out for offside before Birmingham hit the post with a header.