Murray blazed high over the bar from the spot on 35 minutes after going down under a clumsy James Tarkowski challenge.
Things could have been even worse for Albion had keeper Mathew Ryan not denied Burnley striker Chris Wood, when he got through one-on-one 17 minutes from time.
Albion made four changes from their 2-0 midweek defeat at Tottenham as captain Bruno, Solly March, Pascal Gross and Murray came into the starting line-up for Ezequiel Schelotto, Beram Kayal, Jose Izquierdo and Tomer Hemed.
Burnley captain Ben Mee returned to the Clarets side after missing two matches, while Charlie Taylor replaced former Albion loanee Stephen Ward, out with a knee injury, at left-back.
Three other former Brighton players were also in the Burnley squad, with Wood up front and Ashley Barnes and Sam Vokes on the bench.
Lewis Dunk did well to head clear under pressure from Wood early on, before Wood had a header superbly kept out by keeper Mathew Ryan although the whistle had already gone for a foul by the Burnley frontman on Shane Duffy.
At the other end, Albion's first sight of goal came on 13 minutes when Dunk sweetly caught the ball on the volley from Anthony Knockaert's free kick but it thundered off team-mate Duffy and behind for a goal kick.
Murray then headed straight at Burnley stopper Nick Pope moments later but the offside flag was up.
Ryan dived low to his left to push away Johann Berg Gudmundsson's low strike on 20 minutes, before a spell of Brighton pressure.
Knockaert's volley cannoned off a post, Dunk's header was superbly headed off the line by Phil Bardsley and then Albion penalty appeals were waved away when Shane Duffy went down under a Scott Arfield challenge.
Brighton continued to look dangerous going forward and Murray had a fierce shot blocked, before the Seagulls were awarded on a penalty on 35 minutes.
Murray went down under a clumsy James Tarkowski challenge and referee Chris Kavanagh had no hesitation in pointing to the spot. Murray stepped up to take the spot-kick himself but Albion's top-scorer blazed the ball high over the bar into the North Stand.
Burnley full-back Charlie Taylor was booked for a foul on Anthony Knockaert early in the second half, before Wood side-footed a shot straight at Ryan on 48 minutes.
Albion went agonisingly close to taking the lead on 52 minutes when Pope got a fingertip to Knockaert's low shot to push the ball wide and just beyond the reach of the stretching Murray.
Burnley broke straight down the other end from the resulting corner and after Gudmundsson's shot was blocked by Bruno, Tarkowski's header was cleared off the line by Dale Stephens.
Burnley had the ball in the back of the net just past the hour mark when Wood fired home after Ryan twice denied Arfield but the offside flag had been up for a good couple of seconds.
The visitors again had the ball in the goal on 72 minutes but it was ruled out as former Seagulls Barnes pushed Ryan over and received a booking for the foul.
Wood should have given the Clarets the lead with 17 minutes to go as he raced clean through and tried to dink Ryan but the Seagulls keeper stood up to push the ball to safety.
Davy Propper and March sent shots wide in the closing stages as Albion failed to score for the fifth time in six league matches but they remain 13th in the table.
BRIGHTON: Ryan; Bruno, Duffy, Dunk, Bong; Knockaert (Izquierdo 83), Stephens, Propper, March; Gross (Brown 74); Murray (Hemed 63). Subs: Krul, Kayal, Goldson, Schelotto.
BURNLEY: Pope; Taylor, Cork, Tarkowski, Mee, Wood (Vokes 81), Hendrick, Defour (Barnes 68), Gudmundsson, Bardsley, Arfield. Subs: Lindegaard, Lowton, Westwood, Wells, Long.