Cardiff had gone four games without a goal until the England Under-21 international destroyed Wednesday in the space of nine minutes.
The Sheffield side were in trouble from the ninth minute when City right-back Rhys Weston was given space on the right flank to pick out Jason Koumas who pounced from 10 yards to steer his glancing heading wide of on-loan keeper Nicky Weaver.
Cardiff should have increased their lead moments later after Dutch centre-back Glenn Loovens threaded the ball to Jerome, but instead of rounding the advancing Weaver he lashed in a shot from the edge of the area that crashed against the top of the bar.
Jerome missed another chance after Koumas picked him out with a curling cross from the right, but his diving header from six yards struck the foot of the post and as Kevin Cooper charged in, his shot was blocked by Richard Wood.
He made amends on 30 minutes with a carbon copy move. Koumas again provided the cross and Jerome beat his marker at the far post to head home from close range.
Jerome grabbed his second of the night seven minutes before the interval. Midfielder Jeff Whitley's lofted pass fell nicely for the young striker who this time kept his cool to lash the ball past the helpless Weaver from 15 yards.
Graeme Lee saw his header hit the bar before Chris Eagles pulled a goal back for the hosts after 65 minutes. The on-loan Manchester United man cut inside to steer a left-foot shot beyond Neil Alexander from 15 yards.
Substitute Lee Bullen almost grabbed a second in injury-time but his close-range strike was pushed round the post as Wednesday staged spirited fightback.