The game, which contained a trial of video assistant referees (VAR), ended in a 2-1 triumph for the Seagulls, with Dale Stephens also getting on the scoresheet, while Bakary Sako's equaliser was not enough for Palace to push on and grab the win.
The hosts suffered a blow in the opening exchanges when Isaiah Brown limped off the pitch after suffering an apparent knee injury.
The Seagulls managed to cope, though, as they created more chances than their opponents in the first half, one of which resulted in a save from Wayne Hennessey.
At the other end, Sako tried his luck with a shot from a tight angle after cutting inside and getting past his marker, but Tim Krul used his legs to prevent a goal.
Brighton eventually made the breakthrough in the 25th minute, when Dale Stephens netted his first goal since 2016 with a thumping shot from the right side of the box after taking two touches.
The hosts completely dominated the first half, but Palace improved after the break, with a switch in formation as manager Roy Hodgson manoeuvred the team into a 4-4-2 system.
The Eagles set up an exciting finish to the game as Sako nabbed an equaliser after surging into space and delivering a 25-yard low shot past Krul.
As the minutes ticked by, the contest became much more open, and Brighton had the chance to regain their lead when March's cutback to Sam Baldock resulted in the ball smashing off the post.
Baldock was involved again a few minutes later, but he was left with his head in his hands after squandering a perfect opportunity that was handed to him on a plate by Kayal's through-ball.
The striker ended up hitting his shot over the bar, leaving manager Chris Hughton in disbelief, but it was not long before he was celebrating.
Murray ended up being the hero in the 87th minute when he bundled the ball into the net after Uwe Hunemeier knocked it back across goal with a header.
The Seagulls will take on Middlesbrough in the next round.