Swansea were totally dominant in the first period and deservedly took the lead via Jonathan de Guzman on 25 minutes after a cross from Leon Britton.
Wilfried Bony drew a great save from Julian Speroni with a close-range effort as Swansea failed to add to their score.
The hosts were made to pay as Palace improved markedly after the break and substitute striker Murray earned a crucial point for his side with eight minutes remaining after being brought down by Chico Flores, who was sent off.
The offence appeared to take place just outside the box yet referee Mike Dean pointed to the spot and Murray, recently back from a long-term injury, converted to notch his first Premier League goal and leave a bitter taste for Garry Monk's Swans.
Swansea have now gone six games without a win and appeared to fade after a hectic league and cup schedule.
Angel Rangel, Britton and Nathan Dyer all freshened up the starting side that performed admirably in their 3-1 Europa League defeat at Napoli on Thursday.
As has become the norm at home, Swansea enjoyed the majority of possession in the first half, hogging 83 per cent of the ball.
Playing with a compact five across the midfield, Tony Pulis' men seemed content to allow the home side to pass it around on the slick surface. A tame glancing header from Marouane Chamakh was their only chance of note in a limp first-half display.
Boos and whistles of derision accompanied every touch from Tom Ince, who reportedly turned down a loan move to Swansea in January and said before the match that the Welsh team would be unable to handle Palace's play.
Monk was forced to make a change to his side after 22 minutes, bringing on Spaniard Jose Canas to replace injured countryman Pablo Hernandez. His parting gift for the fans was a left-footed strike, dragged narrowly wide.
Pulis also had to make an unwanted substitution when Chamakh also went off injured, making way for Cameron Jerome.
Some crisp interplay from Swansea carved out the opening goal after 25 minutes. Ashley Williams played the ball into Bony, whose lay-off to Britton drew the Palace defence out of position. When de Guzman latched onto Britton's inch-perfect pass, all he had to do was side-foot it past Speroni.
Bony forced a good diving save from Speroni when he met a first-time Rangel cross, after a curling ball from former Palace junior Wayne Routledge.
A cheer rang out around the ground at the start of the second half when it was announced the fans' villain of the piece, Ince, was being replaced by Murray as Pulis switched to a more attacking 4-4-2 formation.
The re-jig got the Eagles back in the game, and they were finally able to give their fans some territory and pressure to cheer, with Yannick Bolasie continually causing problems down the flanks.
Leroy Lita made only his second appearance for Swansea, coming on for the weary-looking Bony with half an hour to go.
A hooked shot from Murray was easily gobbled up by Michel Vorm and a header from Jerome was comfortably saved a minute later.
The game turned after 81 minutes when a mix-up led to a penalty for Palace and a red card for Flores.
Vorm and his defence failed to clear a ball over the top, allowing Murray to toe-poke it past the goalkeeper 20 yards out from goal.
With an open goal beckoning, he was brought down just outside the box, but referee Dean, some 30 yards behind play, pointed to the spot and sent Flores off. Murray fired the penalty into the top-right corner.
He was on the end of the visitors' next attack too, when his goal-bound shot was blocked by replacement Jordi Amat.
In injury time, Palace sub Jerome Thomas tried to persuade Dean to give another spot-kick but was booked for diving as the game ended all-square.