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The home side had hit the woodwork less than 10 minutes into the tie when Sam Byram charged down the flank before eventually getting the ball to Stuart Dallas, but Tom Heaton was there to make the save.
Burnley wanted a penalty when Lukas Jutkiewicz thought that Giuseppe Bellusci had handled the ball, but referee Kevin Friend disagreed, allowing Burnley to test Heaton with a 30-yard effort down the other end moments later.
Both sides continued to look lively at the start of the second half as Vossen was denied by a block from Bellusci before Heaton was needed to halt Chris Wood down the other end.
Leeds were denied by the woodwork again just before the hour mark as Mowatt unleashed a powerful drive from just outside the box, but it glanced the post as it went behind for a goal kick.
Leeds started to look the more likely as the final 20 minutes approached with Byram heading the ball on for Wood, but he wasted the opportunity by nodding it over.
The home side's persistence finally paid off inside the last 10 minutes as Mirco Antenucci was given too much space inside the Burnley box, allowing last season's top scorer to curl the ball in from 20 yards out.
Sean Dyche's men responded quickly though as Tendayi Darikwa delivered the ball into the box for Sam Vokes, who rose high to nod the cross into the back of the net for a swift equaliser, ensuring that the sides finished level.