The 25-year-old smashed home the opening goal on 13 minutes, before doubling the Blues' advantage from the penalty spot shortly after.
The visitors started the brighter and saw their dominance rewarded on 13 minutes, when the Scot unleashed a fierce effort past former Birmingham goalkeeper Jack Butland.
Zigic was then adjudged to have been fouled inside the area, as Caddis stepped up and secured his and team's second of the afternoon from 12 yards.
Play was temporarily halted due to a malfunctioning floodlight but the fluidity of Birmingham's attack never dimmed and it was soon three.
Having won the penalty that enabled Clark's charges to establish daylight over the Tykes, the Serbian then started and finished the attack that led to the team's third.
It would prove as close as the hosts would come to scoring as the West Midlands outfit held out for a comfortable victory and move further clear of the relegation zone in England's second tier.