Clarke reigns against Vic.
Chris Fagan put City ahead with a volley after Jon Newby and Matthew Bailey had failed to make the most of clear-cut chances for the home side.
Newby and Bailey then made amends when they combined and Bailey turned to slam a shot into the roof of the net.
But Northwich, who beat Charlton in round one, were level for less than two minutes after a Bailey mistake.
The defender failed to deal with a long ball and was out-muscled by Clarke, whose shot bobbled in off the far post.
Clarke then latched on to Scott Kerr's pass and found the same corner.