Ok, I took a quick look at the NHL Constitution and by-laws. A team can challenge a penalty that involves the loss of draft picks to the Board of Governors, and the commissioner's punishment can be reversed by a 3/4 vote. The by-laws provide a fairly detailed procedure for a hearing before the Board of Governors. Interestingly, there's a provision that allows the team challenging a penalty to move to disqualify a presiding member for bias, which obviously the Devils could argue is the case since every other team would stand to benefit by the Devils losing a draft pick. This could be a backdoor means of bringing the case before a neutral arbitrator. I don't see anything that provides for a time period after a penalty is announced to bring a challenge.
This is a very quick analysis, so don't take my word as gospel.
Interesting - I still don't see how they get this before an arbitrator, or how the arbitrator rules in the Devils favor, but it could be a way out.