United opened the scoring in the Championship clash at Turf Moor when David Cotterill embarked on a run from inside his own half shortly after half-hour mark.
Cotterill was cynically hacked down by David Unsworth and Beattie netted with a spectacular 30-yard strike from the resulting free-kick.
However, Paddy Kenny was forced into a superb save two minutes later when Kyle Lafferty fired goalwards.
Burnley were punished in the 53rd minute when Cotterill crossed from the left to Sharp, who headed in.
A late rally saw Andy Cole sweep the ball home for Burnley on 80 minutes.
Cole was then denied an equaliser when Kenny superbly parried his close-range volley.