Chris Hughton’s men dominated for long spells but their search for a first Premier League victory since the start of last month goes on after Murray blazed his 36th-minute spot-kick high over the crossbar.
The miss left boss Hughton saying: “We can stay in games, but we need to find the formula to get these goals.”
Hughton made four changes following the midweek defeat to Tottenham.
Murray was one of the men recalled and he headed an early chance straight at keeper Nick Pope.
Johann Berg Gudmundsson’s low shot then forced a rare first-half save out of Brighton keeper Mathew Ryan in the 21st minute before the hosts almost edged ahead.
Anthony Knockaert connected well with Pascal Gross’ delivery, only to see his powerful effort rebound off the right post.
Brighton continued to push and defender Lewis Dunk was unlucky with a header which was nodded off the line by Phil Bardsley.
Burnley boss Sean Dyche appeared to be agitated with decisions going against his team, while the home crowd vented their frustration at perceived time-wasting from the Clarets.
Dyche had reason to be concerned 10 minutes before the break when the Seagulls were given a golden opportunity to grab the goal their dominance deserved with the award of a penalty for James Tarkowski’s challenge on Murray.
Brighton’s five-goal top-scorer stepped up to take the spot-kick but powered his effort over the bar.
Burnley striker Chris Wood, who spent a season on loan with the Seagulls, had an early sight of goal in the second half but Ryan saved comfortably.
Wood did have the ball in the net on the hour mark, after Ryan had twice denied Scott Arfield, only to see the effort ruled out for offside.
Burnley should have secured all three points 17 minutes from time.
Sub Ashley Barnes freed Wood but the New Zealand international’s attempt was blocked by Ryan, ensuring a fourth successive draw between the sides.
Dyche said: “It’s been a tough week, but to get three clean sheets and seven points from three games is very pleasing.”