Healy coolly slotted home a 41st minute penalty after Lee Croft's clumsy challenge had brought down Steve Stone.
It was the perfect present for the Northern Irish marksman as he celebrated his 27th birthday with three valuable points.
Leeds boss Kevin Blackwell handed debuts to loan signings Geoff Horsfield and Tony Warner while summer buys Ian Westlake and Seb Carole had to be content with a place on the bench.
Norwich, who were aiming to reverse a trend that saw them lose 13 away games last season, gave a start to Croft in midfield.
It was Leeds who made the brighter start, with Eirik Bakke and Gary Kelly combining before Kelly's ball found Healy on the edge of the box.
Healy controlled the ball well and spun quickly before unleashing a shot that flew just past the post.
It marked Leeds' intent to adopt a more attacking style of football this season but they struggled to kick on and make the most of their early efforts.
Instead, Norwich worked their way into the tie and Robert Earnshaw could quite easily have had a first-half hat-trick.
His partnership with Darren Huckerby showed frightening pace and that was far too much for Paul Butler to cope with as Matthew Kilgallon tried desperately to cover the entire defensive line.
Tony Warner rescued Leeds from Earnshaw's first attempt, standing tall when the Norwich striker had only the keeper to beat.
Earnshaw's next strike flew wide before a 20-yard effort struck the bottom of the post to send out another warning to the hosts.
Leeds had their own chances but worryingly reverted to the 4-3-3 formation that brought so few goals last year before Healy's penalty broke the deadlock. Shaun Derry saw his effort tipped over by Paul Gallacher before Horsfield fired over from close range after pouncing on a loose ball.
Horsfield then fumbled his shot into the post after the break with the goal at his mercy while Carole could have scored with his first touch after replacing Healy, but it was still Norwich – and particularly Earnshaw – who were the most likely scorers.
But he was denied twice more by Warner and found the post again in the second half as Leeds clung on for the points.