United fell behind after just 59 seconds at Turf Moor as lapse defending allowed room for Scott Arfield to fire home past Marco Silvestri.
George Boyd almost doubled the lead with a carbon copy chance soon after but Leeds were much improved after the interval with Wood squandering two gilt-edged chances directly in front of goal.
Inviting crosses from first Mustapha Carayol and then Charlie Taylor were sent waywardly behind the goal before the Clarets somehow missed an even easier opportunity in injury time when Ashley Barnes fired wide of an open goal.
United have now picked up just one point from their last four games but Cooper believes Wood will erase the memory of his Burnley blunders in tomorrow’s trip to Birmingham City.
“He’ll be hurting,” said Cooper. “He’ll beat himself up but he’s got a chance to make it right and he will do.”
Whites head coach Steve Evans said: “The net is three quarters empty and you’d normally expect him to put them in, but he’ll come back on Tuesday, he’ll lead the line. His performance all round other than that was very good.
“We’ve bossed it with some real quality of passing and movement. The mystery to all of us, is that we’ve lost the game.”
Leeds sit 15th – 11 points clear of third-bottom MK Dons.
Burnley: Heaton, Lowton, Mee, Keane, Ward, Boyd, Jones (Marney 76), Barton, Arfield, Vokes (Barnes 71), Gray (Taylor 89). Unused substitutes: Robinson, Tarkowski, Dyer, Darikwa.
Leeds United: Silvestri, Berardi, Bamba, Cooper, Taylor, Diagouraga (Adeyemi 82), Murphy, Cook, Dallas (Mowatt 87), Carayol (Antenucci 66), Wood. Unused substitutes: Peacock-Farrell, Coyle, Bellusci, Phillips.
Referee: L Mason (Lancashire).