Alex McLeish's Blues had far too much quality for their League Two hosts as they eased to an emphatic victory.
The Premier League newcomers made a flying start as Sebastian Larsson's pass allowed Garry O'Connor to lash home the opener from 20 yards.
Larsson then doubled the Blues' lead just before the half-hour mark, having previously struck the woodwork with a free-kick.
The third goal followed just two minutes later as Kevin Phillips volleyed home to remind McLeish he remains clinical despite his advanced years.
Gloss was put on a polished display when trialist Albert Fataki added a late fourth.