The Royals haven't won on their travels since December at Leicester, but Leeds are on the climb and there was no stopping them.
Marcus Hahnemann, Reading's American international keeper, twice denied Hulse and Aaron Lennon with desperate saves before Healy pounced in the 36th minute.
Reading were guilty of suicidal defending when Ibrahima Sonko lost possession too easily to Lennon who neatly played Healy through.
The Northern Ireland international made ground into the penalty box before coolly clipping the ball past the stranded Hahnemann.
Promotion-chasers Reading showed their calibre in flashes and Jermaine Wright was fortunate that referee Chris Foy ruled that his penalty area challenge on striker Nicky Forster was fair.
Forster should have done better when Leeds defender Clarke Carlisle allowed him to wriggle free on the right, the shot zipping outside Neil Sullivan's far post.
Steven Sidwell and Icelander Ingimarsson squandered good first-half chances for the Royals.
Les Ferdinand claimed Reading should have had a penalty two minutes into the second half when he was shoved by Sean Gregan but the referee ruled that the veteran striker had handled.
Ferdinand was also guilty of a bad miss, volleying over the bar from the edge of the six yard box when Sullivan could only palm Glen Little's cross to him.
Hulse then put the game beyond Reading's reach, firing past Hahnemann from 19 yards in the 57th and 63rd minutes, the striker earning a yellow card for removing his shirt amid wild celebrations after his second goal.
Substitute Lloyd Owusu denied Leeds a clean sheet with a neat back-heel from an acute angle in the 89th minute but Leeds were worthy winners, though Sullivan had to be alert to save a header from his own defender Gary Kelly just before Owusu's goal.
Hulse deservedly left the field to a deafening ovation from Leeds fans when he was substituted 19 minutes from the end.