After a goalless first half, Gary Hooper struck on the rebound to put the hosts in front.
A mistake by Cardiff goalkeeper David Marshall led to an own goal from Lee Peltier, before Hooper scored his second in added time.
Sixth-placed Wednesday now have an unassailable seven-point lead over Cardiff in seventh with one game left.
Carlos Carvalhal's side will have to wait until the final weekend - when they are away at Wolves - to discover their play-off opponents.
They are guaranteed to finish at least sixth but could yet climb to fifth, with current occupants Derby at third-placed Brighton on Monday.
Wednesday only needed a draw against Cardiff to secure the final play-off spot but looked intent on claiming victory with a forceful start.