The Bournemouth striker then produced a stunning overhead kick on his own goalline to prevent Jason Shackell's 23rd-minute header from finding the back of the net.
Sam Vokes broke away from the Bournemouth defence after half an hour and bore down on goal, but Steve Cook recovered well to thwart the Burnley striker.
Vokes had another chance to open the scoring five minutes later after Lee Camp palmed Dean Marney's powerful effort into the path of the striker, but 24-year-old could not make the most of the chance.
Just five minutes after the break, Tokelo Rantie struck a powerful volley beyond the reach of Heaton to give the away side the lead.
Substitute Keith Treacy and Vokes were both denied by Camp after an hour, and Danny Ings saw his appeals for a penalty waved away as Burnley pinned Bournemouth inside their own half.
However, with six minutes remaining, Ings beat Camp to the ball to nod home to send the Turf Moor faithful into raptures.