Healy, celebrating his 27th birthday, scored in the 41st minute after an ill-advised challenge on Eirik Bakke in the area by Norwich's Lee Croft.
Norwich paid the price for failing to take their chances as striker Robert Earnshaw hit the post twice and missed three more decent opportunities.
Leeds' Geoff Horsfield saw his second-half effort drift just wide.
The Canaries made the better start, with Leeds keeper Tony Warner forced into early service to keep the lively Earnshaw at bay.
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Healy was also looking sharp and went close with an early strike after some clever play from Shaun Derry, Gary Kelly and Horsfield.
Norwich's Darren Huckerby latched onto a loose Derry pass and set up Earnsha, but the Welshman saw his shot saved by Warner.
Derry's well-struck volley was turned onto the crossbar by Norwich keeper Paul Gallacher, who also saved well from Healy.
Leeds finished the stronger of the two teams, but could not find the second goal they were looking for.
Leeds boss Kevin Blackwell:
"We're up and running now. Whether it exorcises the ghost completely I don't know, that remains to be seen.
"But what it does do is let the lads know that if they start feeling sorry for themselves it will be a very long and hard season."
"The only downsides were that we didn't take our chances - and the penalty. I thought it was a little bit harsh.
"Other than that I couldn't have asked for any more from my players."
Leeds: Warner, Kelly, Crainey (Moore 85), Butler, Kilgallon, Lewis, Stone, Bakke (Westlake 77), Derry, Healy (Carole 69), Horsfield.
Subs Not Used: Gregan, Blake.
Booked: Healy, Butler.
Goals: Healy 41 pen.
Norwich: Gallacher, Drury, Shackell, Doherty, Colin (Thorne 82), Croft (McKenzie 56), Safri, Etuhu, Robinson (McVeigh 71), Huckerby, Earnshaw.
Subs Not Used: Fleming, Lewis.
Att: 22,417.
Ref: G Laws (Tyne & Wear).