Palace went into the game having lost all 13 Premier League matches in which they had conceded first this season and they had to do things the hard way again after Bradley Johnson gave Norwich a 39th-minute lead.
Things should have got even tougher for the hosts before half-time as Marouane Chamakh raised his hand to Wes Hoolahan's face, but the Moroccan only received a yellow card.
Palace made the most of that let-off, as they headed into the break level when Puncheon converted from the spot after Mile Jedinak had been fouled.
The second half saw the home side generally dominate, but after Cameron Jerome had missed a good chance to put them ahead, they struggled to find a way through Norwich's resolute defence.
Palace were unlucky not to get another penalty late on before Leroy Fer was dismissed for a second bookable offence, but the hosts could not make the most of their man advantage.
Both managers made two changes from their respective narrow defeats at the hands of the Manchester clubs last time out, with Kagisho Dikgacoi and Chamakh returning for Palace, while Michael Turner and Steven Whittaker came back into the Norwich defence.
After a slow start, an error by Norwich left-back Martin Olsson in the 30th minute presented Palace with a chance, as Puncheon stole possession and crossed to the back post, but Jerome fired over from close range.
Norwich gradually grew into the match and six minutes before the break they made Palace pay for not capitalising on their superiority. Gary Hooper chased down a long ball and cut a pass back to the edge of the area for Johnson, who clipped a shot in off the right-hand post.
A few moments later, Chamakh escaped being sent off for pushing Hoolahan in the face and Norwich were reeling soon after when Fer conceded a penalty for pushing Jedinak, with Puncheon burying the resulting spot-kick.
Both sides created good chances early in the second half.
First, Nathan Redmond forced Julian Speroni into a good stop from a tight angle before Jerome blasted over after Norwich had failed to clear a corner.
Palace were well on top as the match entered its final half-hour, but Pulis’ men struggled to break down the visitors' defence.
Norwich did fashion a rare chance in the 68th minute when Hooper controlled the ball and turned on the edge of the area, before Speroni got down to make a crucial save from his shot.
The home side felt aggrieved in the final stages when Johnson appeared to handle in the area, only for the referee to wave away the protests.
Palace were given a boost when Fer received a second yellow card for a strong tackle on Jedinak, but they could not make their man advantage count.
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