I pulled all the data from 2005-06 through 2012-13. So I could include last season I looked at points-per-game rather than just points. Here's a scatter-plot of Regular Season Points per Game versus PDO.
The correlation is 0.6448986, which is highly significant (p-value < 2e-16).
Taken by themselves, both Shooting Percentage and Save Percentage are predictive of Regular Season success. The correlation for Shooting Percentage is 0.332574 and the correlation for Save Percentage is 0.480705.
Regular Season PDO and Playoff Success
I looked at Playoff success two ways. One way was win percentage. The other was to assign a score to progression through each round. A team that lost in the first round got a score of 1. A team that lost in the second round got a score of 2 and so forth. The Stanley Cup Champions got a score of 5. I looked at Regular Season PDO, Regular Season Shooting Percentage, and Regular Season Save Percentage as a predictors of Playoff success.
Regular Season PDO is minimally predictive of Playoff success. If you look at Playoff Win Percentage, the correlation is 0.142 (p-value 0.11). If you look at progression score, the correlation is 0.143 (p-value 0.11).
Regular Season Shooting Percentage is not predictive of Playoff success. If you look at Playoff Win Percentage, the correlation is 0.008 (p-value 0.93). If you look at progression score, the correlation is 0.066 (p-value 0.46).
Regular Season Save Percentage is minimally predictive of Playoff success. If you look at Playoff Win Percentage, the correlation is 0.157 (p-value 0.077). If you look at progression score, the correlation is 0.100 (p-value 0.26).
So the shooting (luck) part of PDO seems to lose it's power completely once you get to the playoffs. Even though the goalie (skill) part of PDO persists somewhat into the playoffs, it only explains about 2.5% of the total variability seen.