Owls keeper Brad Jones was red carded on 67 minutes for bringing down Healy as he raced clear and the Northern Ireland international striker tucked away the spot-kick himself.
Wednesday, looking to bounce back after their 4-1 Carling Cup defeat at the hands of lowly Wrexham in midweek, survived an early scare when Eddie Lewis was toppled by Frank Simek on the edge of the area as the pair chased a long ball downfield. However, Healy went for power and blasted the free-kick against the Owls defence.
New signing Lloyd Sam, making his debut after his loan signing from Charlton Athletic, carved out an opening for the Hillsborough side, after cleverly weaving his way past Stephen Crainey.
The youngster unselfishly picked out the unmarked Deon Burton but, from ten yards out, the striker, back after injury, swung his right foot and miskicked.
Tony Warner produced a stunning one-handed save as Wednesday looked for an early breakthrough, diving to his right to palm away Chris Brunt's stinging 25-yard free-kick before the winger was foiled again by the Leeds keeper as he spread himself to scoop up Brunt's close-range shot.
Leeds, for all their possession and several promising set-pieces, couldn't force a save from Jones, while Wednesday were content to catch the visitors on the break.
The lively Brunt pulled another long-range strike wide of the target before Sam almost snatched the opener on 38 minutes.
The teenager, in space on the right flank, crossed from the tightest of angles. The ball cannoned off the far post and Burton was the first to react but, from only six yards out, he scooped the ball over the top.
Leeds were lucky to survive in first-half stoppage time when Sam carved his way into the area only to mis-kick. The ball span to midfielder Burton O'Brien whose low drive took a deflection and, with Warner wrong-footed, went just past the post.
Sam, who was proving a handful for the Leeds back four, then almost found a way through with a shot on the turn that was smothered on the line by the Leeds keeper before the visitors nearly broke the deadlock in the 55th minute.
Jones was forced to get down quickly to push away Seb Carole's low strike and, when Lewis hammered in a low drive, it was blocked by Simek eight yards out.
Jones was red carded for a professional foul on 67 minutes when he brought down Healy as he raced through. Reserve keeper Chris Adamson came on for John Hills but was beaten for power as Healy slammed the ball into the bottom left-hand corner of the net.
It proved to be the decisive goal as United hung on to take three vital points.