Referee Alan Wiley broke Manchester City's hearts by sending off defender Daniel Van Buyten and pointing to the spot with only 14 minutes left.
City, much the more accomplished side, conceded a 23rd minute goal to Stephen McPhail's curling free-kick, but battled back to equalise with Nicolas Anelka's 21st goal of the season.
But with a draw looking ever more likely, Alan Smith hit the deck on the fringe of the penalty area, even though Van Buyten appeared to make no contact, and Wiley awarded the spot-kick, compounding the felony by showing the Belgian a red card.
Viduka sent David James the wrong way, lashing the ball just inside the keeper's right post, to the Leeds crowd's immense relief.
Viduka claimed he should have had a penalty as early as the tenth minute for Richard Dunne's clumsy challenge, but significantly there were no appeals from the Leeds striker's colleagues.
City committed several early fouls and paid for their indiscipline when Leeds took the lead. Michael Tarnat conceded a free-kick by bringing down Jermaine Pennant just outside the 18-yard box on the right.
McPhail, only in the side because Eirik Bakke was still injured, curled the free-kick in off the far post, with Smith's leap obviously distracting James.
However, out of sorts Leeds found it hard to build on their encouraging start, City dictating long phases of play.
Tarnat almost atoned for his part in the Leeds opener when he crashed a 30-yard shot against Robinson's left post and Anelka should have done better with a tame shot on the turn.
City were easily the more cohesive side and Robbie Fowler almost scored against his old club with a deflected drive which dipped only just over the bar.
City had lacked the killer instinct they showed to defeat Manchester United 4-1 in their last match, but they recaptured it to equalise a minute before the break.
Neat inter-passing ended with Joey Barton and Antoine Sibierski playing Anelka through to drive low into the left corner from just outside the box.
Robinson pulled off the save of the match to keep out Sibierski's header a minute after the break and it proved crucial.
So, a great start for Leeds' new owners, and United are now on the coat tails of both Portsmouth and Leicester with everything to play for.