Gifted the chance to score just his side's second goal in six Premier League matches, Murray wastefully fired into the stands as the home side had to settle for a point and the end of a three-match losing run.
Brighton, without a win in six league outings, defied their concerning run of form in an entertaining opening 45 minutes that somehow failed to produce a goal.
Murray should have calmed the nerves from the spot when he tangled with James Tarkowski in the 36th minute, only to wildly hit his attempt over the crossbar.
Burnley improved after the interval and might have snatched three points but for the intervention of Mat Ryan, who produced a double save from Scott Arfield and later denied Chris Wood in a one-on-one situation.
While the inability to take those chances means Sean Dyche's side lose ground in the fight for an unlikely top-four spot, their lofty top-half position will still be looked upon with envy by Chris Hughton's men as they battle to arrest their slide.
FT: Albion claim another point at the Amex but will feel unfortunate to not claim all three. Murray fired a penalty over the bar in the first half and Knockaert also struck the post. Positive display but the Seagulls needed a better end product. #BHAFC 0 #BFC 0 #BHABUR pic.twitter.com/GyZJo2iQ9a — Brighton & Hove Albion (@OfficialBHAFC) December 16, 2017
Brighton commanded possession over the opening exchanges and began to probe with purpose by the 15th minute.
Lewis Dunk's promising volley was inadvertently blocked by team-mate Shane Duffy, before Murray headed straight at Nick Pope from Bruno's teasing cross.
They soon went even closer to the opener. After Ryan dealt with Johann Gudmundsson's effort, the hosts raced up the other end and saw Anthony Knockaert turn Pascal Gross' cut-back against the near-post.
And the Clarets appeared to be hanging on when Phil Bardsley was required to nod Dunk's goal-bound header off the line.
The sustained pressure should have resulted in a lead shortly after the half-hour. Tarkowski clumsily bundled Murray over inside the area, but the striker poorly blasted over the bar from the resultant spot-kick.
Burnley set about asserting more authority after the restart with Wood, holding his place ahead of midweek match-winner Ashley Barnes, side-footing a tame finish at Ryan on the counter.
The goalkeeper had more work to do on the hour, producing a smart double save at close-range to keep out Arfield's hurried pair of left-footed efforts.
And the Australian shot-stopper came to his team's rescue once more with just over 15 minutes remaining, racing off his line and remaining big to smother a golden one-on-one opportunity for Wood following substitute Ashley Barnes' through-ball.
Key Opta stats:
- Brighton have picked up just three points in their last seven Premier League games (D3 L4), after accruing 11 in the seven before that (W3 D2 L2).
- Burnley’s Premier League games this season have produced a divisional-low 28 goals this season (16 for, 12 against).
- The Seagulls have netted just one goal in their last six Premier League games, after scoring eight in the six before that.
- Only Manchester United (10) have kept more Premier League clean sheets this season than Burnley (9, level with Arsenal).
- The Clarets have only made 15 changes to their starting line-up in the Premier League this season, nine fewer than any other side.
- Glenn Murray has now missed three of his last five Premier League penalties.