Jack Harrison’s deft finish on 65 minutes was all that separated the two sides at Elland Road.
The Whites had been denied by two wonder saves from visiting goalkeeper Keiren Westwood during a dominant first half.
United were also left counting the cost of some wayward finishing, with Patrick Bamford in particular having an off-day.
But the relentless pressure finally paid off when Harrison fired home.
Westwood had to be at his very best to keep out a Tyler Roberts header after 13 minutes as the home side started to find their stride.
Gary Hooper could only hit the side-netting before Adam Reach had a shot blocked minutes later as Kiko Casilla failed to deal with Barry Bannan’s free-kick.
Egzjan Alioski blazed over under pressure from Westwood, before Roberts flashed an effort inches wide.
Casilla pushed Bannan’s curling effort around the post with a brilliant one-handed save in the 25th minute.