The hosts began brightly as they put some early pressure on their opponents, but the Seagulls went on to carve out the first real opportunity of the game.
Nonetheless, Burnley worked their way into the game and enjoyed several bright spells, albeit without troubling Brighton too much in the final third.
Burnley also had two penalty appeals for handball rejected in the first half, with Sean Dyche's men still yet to be awarded a spot kick in the Premier League this season.
In the second half, Chris Wood had to go off for the hosts, blunting their attack, while a Gross effort that Pope dealt with easily was the closest that Brighton came to a winner.
The result means that Burnley need just one more point from their last two games to secure their place in the Europa League, while Brighton are now five points clear of the relegation zone with three matches to play.
BURNLEY (4-4-2): Pope; Lowton, Tarkowski, Long, Ward; Lennon (N'Koudou 72'), Cork, Westwood (Hendrick 81'), Gudmundsson; Barnes, Wood (Vokes 66').
BRIGHTON & HOVE ALBION (4-2-3-1): Ryan; Bruno (Schelotto 89'), Duffy, Dunk, Bong; Stephens, Kayal; Knockaert, Gross, Izquierdo (March 85'); Ulloa (Murray 67')