Jason Koumas put the visitors ahead on nine minutes with a brave header from Rhys Weston's right-wing cross.
Jerome hit the bar and post before heading in Koumas' cross and drilling home Jeff Whitley's through-ball in the space of nine minutes.
Chris Eagles responded for Wednesday with a fine left-foot effort and Richie Partridge was denied by a fine save.
Cardiff boss Dave Jones:
"The injuries to Kevin Cooper and Jason Koumas will test us a little bit because we are only running with a 19-man squad and two of those are keepers.
"But it's a nice feeling to be fifth and I'd rather be where I am than be down the bottom end.
"We attacked them, we didn't sit back and I thought it was a good, high-tempo 45 minutes which has got us the win."
Sheff Wed: Weaver, Heckingbottom, Lee, Wood, Simek, Whelan, O'Brien, Brunt, Eagles (Partridge 80), Corr (Bullen 61), Graham (Agbonlahor 80).
Subs Not Used: Adamson, Rocastle.
Booked: Simek.
Cardiff: Alexander, Barker, Purse, Weston, Loovens, Whitley, Koumas, Ledley, Cooper (Parry 59), Lee (Ardley 88), Jerome.
Subs Not Used: Margetson, Cox, Ricketts.
Booked: Whitley, Jerome.
Att: 20,324
Ref: T Kettle (Berkshire).