Jay Emmanuel-Thomas was behind most of City’s best first-half moves. His jinking run and low cross set up Joe Bryan to shoot the hosts ahead from close range in the eighth minute.
Smith, on loan from Fulham, doubled City’s advantage just before the break with a similar finish after Luke Freeman had got behind the County defence to latch on to Wade Elliott’s chip.
Emmanuel-Thomas settled any doubts about the outcome in the 62nd minute, taking a touch in a crowded penalty area and creating space to shoot low past Roy Carroll from 15 yards.
Centre-back Derrick Williams completed an emphatic victory with a cool low finish from Bryan’s cross six minutes from time.
Carroll made good saves to deny Mark Little and Bryan, while disappointing County barely tested home goalkeeper Frank Fielding and were largely outplayed.