The game was short on quality in the opening 25 minutes, with the only piece of action being a booking for Stephen Warnock for a foul on Woolford.
However, the home side came close to taking the lead in the 28th minute when Martyn Waghorn's effort beat Paddy Kenny, but not the post, and Woolford blazed his subsequent effort of the bar.
Leeds's best chance fell to Jason Pearce, but the central defender clumsily hooked the ball over the bar after Ross McCormack set up his teammate with a cushioned header.
Woolford broke the deadlock in the 65th minute after a counter-attack by the home side. Waghorn and Dave Martin combined well and the latter released the 27-year-old striker, who struck the ball into the bottom corner from the edge of the area.