Much like their giantkilling of Charlton in the last round, Northwich roared out of the blocks and wasted an early opportunity to take the lead when Jon Newby fired wide.
And Andy Preece's side were made to pay for that miss after 18 minutes when Fagan fired a fantastic left-foot volley past Curtis Aspden. Undeterred the Vics continued to press and Lee Elam might have levelled proceedings but he dragged his shot wide.
Bailey deservedly drew the home side level just two minutes after the break when the big centre-half swivelled smartly in the box to fire a volley past Robert Burch. But Bailey turned from hero to zero a minute later when he was caught napping by Clarke and the 21-year-old darted clear to fire the Imps ahead.
And when Clarke notched his second in the 63rd minute - again latching onto a Scott Kerr throughball before smashing home from the edge of the box - Northwich's giantkilling hopes were over.