Dan Lomas put his side ahead from the penalty spot before Smith doubled City’s lead before the break.
Ryan Harkin pulled a goal back for United before Smith secured City’s passage controlling the ball before firing home.
They face Hellenic Premier leaders Worcester City in the next round, who left it late to secure a 2-0 victory at Boston Town.
Dylan Hart opened the scoring in the 90th minute before Kyle Belmonte sealed their passage.
Great Wakering Rovers reached the fourth round for their first time in 22 years after a 3-0 win over Fakenham Town.
Callum Boylan, Harry Talbot, and Aiden Brown all scored as the Essex Senior League side progressed.
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CRAIG'S BOOSTS
MANAGERIAL stalwart Craig Edwards is back in charge of Barking – 23 years after he left!
LENNIE GETS THE LOVE
VETERAN caretaker chief Lennie Lawrence says he is “excited” to be handed the reins at Hartlepool United permanently – just a few weeks before his 77th birthday!
MOTORS FEELS LIKE HOLMES!
WHEN he was younger, Danny Holmes played video games that simulated being a football manager.
REES HITS FAB FOUR AS TOWN ROMP IT
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DULWICH DISPLAY DELIGHTS COACHES
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ANCHORS A RAPID
ASHTON UNITED hit two goals in two minutes to come from behind to win at Stockton Town.
SEAGULLS KEEP THE HEAT ON ROMANS
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CLINICAL COLEMAN KEEN AS MUSTARD
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KEV SEES RED IN REE PANT!
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MARVEL MATTY DENIES MOORS
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