Seasiders defender Alex Baptiste opened the scoring four minutes before the break in the Championship clash when he fired home a cross from Barry Ferguson.
Yet the high-flying Eagles ensured their fine start to the season continued as Murray - a 70th-minute replacement for Darren Ambrose - levelled the scores with 11 minutes remaining.
Blackpool went close inside the third minute when veteran striker Brett Ormerod hit a 12-yard shot which drew a fine parried save from Palace goalkeeper Julian Speroni.
Two minutes later, Tangerines midfielder Keith Southern drilled a 25-yard free-kick which Palace defender Patrick McCarthy did well to block.
The visitors continued to probe and went close again in the ninth minute when Ormerod's close-range strike was blocked by Dean Moxey.
Remarkably, Blackpool fashioned yet another opening in the 12th minute when striker Billy Clarke hit a 20-yard shot which Speroni did well to parry.
With 20 minutes played, evergreen frontman Kevin Phillips saw a low right-footed shot flash just wide of the far post.
Midway through the opening period, Palace created their first opening of the half.
Ambrose crossed dangerously and defender Craig Cathcart cleared the ball only as far as Ryan McGivern but he could only steer his header wide.
In the 26th minute, Blackpool left-back Stephen Crainey fouled Jermaine Easter and Eagles midfielder Ambrose delivered a dangerous free-kick from the right flank.
McCarthy arrived to meet the ball with a firm header which drew a brilliant save from Seasiders goalkeeper Matt Gilks.
Shortly before the half-hour mark, defender McGivern crossed well only for Easter to send a poor header weakly wide.
Two minutes later, Gary Taylor-Fletcher volleyed a fierce 30-yard effort over the bar.
Ambrose then fired over from distance before Baptiste opened the scoring when he swept home a right-footed shot from six yards after connecting sweetly with Ferguson's cross.
Ten minutes after the restart, McCarthy almost levelled when he headed straight at Gilks from Ambrose's corner.
In the 67th minute, Phillips produced a weak shot from 12 yards when he should have done better and Moxey then hit the bar from distance.
Murray then entered the action and soon had a header saved by Gilks before he equalised in the 79th minute.
Moxey crossed from the left flank and Murray arrived to sweep the ball home right-footed.
Southern and Murray then both went close from distance but neither side could find a winner.