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Wednesday's Finnish international Shefki Kuqi steered a tame header straight at keeper Steve Banks before almost snatching the lead in the sixth minute.
Midfielder Alan Quinn picked out the Wednesday striker with a cross from the left flank and Kuqi's fierce header crashed against the bar with Banks beaten.
Simon Donnelly missed another opportunity as he charged into the area but from 12 yards he slammed the ball straight at Banks before the unpredictable Sibon grabbed the breakthrough in the 34th minute.
The lanky Dutchman, in space on the right, had options to pass with Kuqi running through unmarked as he lashed a stinging shot from 20 yards into the far corner.
Wednesday were in trouble three minutes before the interval, when defender Danny Maddix, who had already been booked, was sent off for a goalmouth skirmish with Ashley Ward as Bradford hit back in the 44th minute.
A Wayne Jacobs cross was met by Andy Gray and when his shot was blocked, Stephen Warnock stuck out a foot to loop the ball over Kevin Pressman.
Ten-man Wednesday ran themselves into the ground in the second half but could not find a way through.
Sibon hit the winner from the penalty spot in the 84th minute after substitute Lloyd Owusu was bundled over in the area by Robert Molenaar.