The visitors were in desperate need for three points after Cardiff City's win over Southampton saw them drop into the bottom three.
It was evenly contested affair in the early stages, although Burnley went close to pulling ahead on 16 minutes when Brighton's Shane Duffy deflected a cross onto his own crossbar.
The visitors 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.
Brighton pushed for an equaliser in the remaining stages of the half, but Burnley had goalkeeper Tom Heaton to thank for the numerous saves before the interval.
While the home side threw everything forward in the second, Burnely managed to bag second on the counter through Woods' clinical finish on 61 minutes.
Bunrley 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.