Josh Brownhill sent Vincent Kompany's side ahead in the 75th minute, but Muric scored an own goal four minutes later, as the points were shared in the key encounter.
The result has left Burnley 19th in the table, seven points from the safety of 17th position, while Brighton are 10th, one point behind ninth-placed Chelsea.
Wilson Odobert had an early shout for a penalty when he went down inside the Brighton box, but the referee was not interested and VAR agreed, much to the frustration of the hosts.
Brighton's first sight of the home side's goal came in the 15th minute through Jakub Moder, but he could only head over the crossbar from a dangerous position.
Burnley had a huge chance to make the breakthrough in the 18th minute through Jacob Bruun Larsen after a smart team move, but the attacker fired wide of the target from close range.
Brighton were back on the attack shortly after, and Burnley goalkeeper Muric had to be alert to keep out a powerful strike from the dangerous Simon Adingra.
Adingra had another opportunity in the 29th minute, but he missed the target after some poor defending from Burnley, with the Seagulls settling into the match after a below-par start at Turf Moor.
Odobert was Burnley's biggest threat, and he fired wide of the post from outside the Brighton box in the 32nd minute, with the score remaining goalless heading towards the end of the first period.
Joao Pedro had a half-chance in the closing stages of the half, but the forward's header from a Pascal Gross delivery was always moving wide of the target, before Muric produced a brilliant save to keep out a free kick from Moder down the other end of the field.
Burnley had a half-chance early in the second period, but Sander Berge's effort from the edge of the box never threatened to cause the Brighton goalkeeper an issue.
David Datro Fofana then somehow headed a cross wide of the post from close range, with a huge chance being wasted by the relegation-threatened hosts.
Brighton had two chances in the period that followed through Gross, but the midfielder saw his attempt from distance deflect just wide, before Muric made a smart stop to keep out his second effort.
Burnley finally made the breakthrough in the 74th minute, with Brownhill on the scoresheet, with the ball rebounding in off the midfielder following a mistake from Brighton's Carlos Baleba.
However, the score was level just five minutes later courtesy of an incredible own goal from Muric, with the goalkeeper allowing the ball to run under his feet.
Burnley had a chance to win it late on through Odobert, but the attacker missed the target, with the points being shared following two mistakes from both sides in the latter stages.