The visitors were in desperate need for three points after Cardiff City's win over Southampton saw them drop into the bottom three.
Burnley got the breakthrough on 25 minutes as Chris Woods broke clean through on goal before firing past a helpless Matt Ryan from 12-yards out.
While the home side threw everything forward in the second, Burnley managed to bag second on the counter through Woods' clinical finish on 61 minutes.
Burnley were then awarded a penalty on 72 minutes after Ashley Barnes was brought down by Ryan. The striker dusted himself off and calmly slotted the spot-kick into the bottom corner.
Brighton pulled a goal back three minutes later through Duffy's close range header following a set-piece from Anthony Knockeart.
The goal proved to be nothing more than a consolation as Sean Dyche's men held on for the 3-1 win to move out of the relegation zone.