The Blues played a very smart if unscrupulous series in that they read and exploited the mood of the public and the NHL administration very well, and they mostly played to their strengths and within their limitations.
After the wins by the Sox and the Pats, on top of numerous other successes in the last 15 years, most Americans were sick of Boston winning things. Nothing personal or malicious for the most part I think, just that repeat triumphs by the same city quickly grows tiresome unless you're from that city. On top of that, the Bs are still carrying their 'Big Bad Bruins' reputation. Obviously this current team has very little, probably too little, that is either big or bad, but these things tend to stick long after the reality that spawned them has gone. Nothing demonstrated these circumstances better than the blatantly, openly biased coverage of the NBC, which played out virtually without criticism outside of Boston.
These two factors allowed the Blues to play dirty and the heavy, delaying style that suited them, and get the refs on their side, because there was almost no care factor or scrutiny as to how St Louis was taking it to Boston as long as they did it, and little willingness to believe that a team with the past rep of the Bruins would ever be on the rough side of an opponent breaking the rules and getting away with it. The Acciari non-call pushed that blindness right to the edge of breaking point, but didn't quite snap it. Otherwise the Blues used their underdog and 'not Boston' cards very well in closing the talent gap between the two squads, and very few people either twigged or cared that they were taking advantage of them.
For the NHL's part, I honesty don't think they actively wanted the Blues to win, they simply wanted a long and interesting series that would maximise air time and revenue. So they were more than happy to change the refereeing after game 3 when it looked like Boston was on the edge of breaking the series open and finishing it quickly, and aware that they could get away with it for the reasons outlined above, but they were also willing to pull the refs back to neutral after game 5 and help send the series to its ultimate game 7 decider. Again, the egregiousness of the Acciari call probably contributed to this in that they realised that things were going too far, but otherwise in terms of meeting their aims the league played things just about perfectly. Of course they shouldn't be involved in manipulating a series for ANY reason, but it seems pretty clear that in this case they did.