The former Southampton winger opened the scoring with a low drive just after the half-hour mark, after good work from boyhood Cardiff fan Joe Ledley.
Ledley himself doubled his new side's lead, tapping in after a Marouane Chamakh header was saved.
And Puncheon, who was a real livewire all afternoon, curled in a sublime third to emphatically seal victory for the Eagles.
The result sees Cardiff plunged further into the relegation mire, with survival looking a tall order, but Palace have done their own chances of beating the drop a world of good after what was a comfortable win.
It was Palace's second successive victory, arriving just seven days after they toppled title-chasing Chelsea, and confirmed a stunning continued revival under their Welsh-born manager Tony Pulis.
Both teams struggled to make headway during an opening 20 minutes notable only for one shot blazed over the bar by Palace's Kagisho Dikgacoi and Cardiff's inability to keep possession in midfield.
A routine Julian Speroni save from a Fraizer Campbell header was the only real goalmouth action, Palace opened the scoring after a fine move.
Local-lad Ledley found space in the box, and waited patiently to pick the right pass, finding Puncheon with room to maneuver, who then shifted the ball quickly and fired home through a crowd of players, to leave Bluebirds goalkeeper David Marshall stranded.
The Bluebirds began to up the tempo after going behind, but the Eagles' rearguard stood firm, and were rarely troubled by the Welshmen.
After the break the home side side did come out with some added impetus, causing some problems from long balls into the box.
Kenwyne Jones beat Speroni with a header, but it was scrambled clear by several bodies in the Palace six-yard box.
The first switch came after 57 minutes when Craig Noone, who had missed the last three games due to a groin injury, replaced a misfiring Wilfried Zaha, while Pulis sent on Marouane Chamakh instead of Yannick Bolasie.
Noone injected some pace and purpose into Cardiff's play, and they looked a far more threatening outfit with him making angled runs from deep, ensuring that Palace's defenders came under sustained pressure for the first time.
But Palace weathered the storm, and grew in composure to dominate the closing stages.
For all of Cardiff's pressing, there was never a sense that a goal was coming, and if anyone was going to add to the scoreline, it would be the visitors, and that proved to be the case.
After a whipped free-kick from Puncheon, substitute Chamakh saw his header well saved by Marshall, but he could only push the ball into the path of Ledley, who tapped into the empty net. There was a muted celebration from the Cardiff fan.
Campbell blazed a close-range header wide late on before Puncheon made sure of the win, curling a wonderful effort past Marshall from 25 yards, to put the gloss on a fine Palace performance, which completed an Eagles double over the hapless Bluebirds.