Bielsa opted to start with the same line-up that earned a late point at Preston North End in midweek as captain Liam Cooper returned to the bench in place of Barry Douglas.
The Yorkshire derby got off to an even start as driving rain filled the Sheffield air.
It was Leeds who were forced into the early defensive action as Gjanni Alioksi and Ben White both produced last ditch tackles in the final third to keep the hosts at bay.
Liam Palmer then tested Kiko Casilla from distance before Luke Ayling swiped and missed from a United corner in what was a slow affair.
White was also involved in the opposition box as he fell under a challenge from Atdhe Nuhiu who appeared to nudge into the defender but referee Tim Robinson waved the penalty appeals away.
The half appeared to be coming to a tame end but a flurry of action forced both goalkeepers into saves.
First Casilla denied Steven Fletcher as he tipped a fierce volley over the bar before Patrick Bamford's header back across goal was tipped inches past the post by Keiran Westwood.
It was to be Bamford's last action of the afternoon as Eddie Nketiah was brought on at the interval by Bielsa.
Leeds flashed an early warning shot as Garry Monk's side as Kalvin Phillips saw an effort fly inches wide of the post.
It was Wednesday, though, who really should have taken the lead as Fletcher smashes the crossbar from inside the box after good work from Danny Fox before missing what appeared to be an open goal.
Barry Bannan opened up the United defence on the left and delivered to find the striker all alone at the back post but he failed to work a shot on target.
Bielsa's side mustered a response as Nketiah drilled low at Westwood before the Argentine made his second change of the afternoon as Liam Cooper replaced Costa.
The change gave Leeds life as a Jack Harrison effort was cleared off the line by Fox before Alioski saw a header crash off the inside of the post as the ball was flung back into the box.
The chance rattled the hosts into life as Casilla was again forced into action to keep the scores level.
He first tipped a Kadeem Harris cross wide before pushing a Nuhiu header away from the target from the resulting corner.
Despite the late flurry of chances neither side could break the deadlock as the Yorkshire derby spoils were shared at Hillsborough.
Sheffield Wednesday: Westwood, Palmer, Fox, Harris, Fletcher, Bannan, Pelupessy (Lee 84'), Nuhiu (Forestieri 86'), Reach, Hutchinson, Iorfa. Subs: Dawsom, Lee Winnall, Murphy, Odubajo.
Leeds United: Casilla, Ayling, Berardi, White, Alioski, Phillips, Klich, Dallas, Costa (Cooper 76'), Harrison, Bamford (Nketiah 45'). Subs: Miazek, Davis, Gotts, Clarke, Roberts.
Referee: Tim Robinson