In order to get the lead teams did not need to significantly outshoot their opposition, at least in the first period. It may be, of course, that teams got their leads in the second when they had more success outshooting. Nonetheless, this is one piece of evidence that may suggest that outshooting does not necessarily result in outscoring.
Once teams have the lead in the third period they have a tendency to sit back. This is what most observers of the game would intuit and so I think this passes the smell test. It also, to a degree, answers the question of how much: teams that are trailing increase their shot total by 11.3%, teams that are leading decrease their shot total by 21.1% and the overall shots in games where one team has the lead decrease by 5.2%. Given that these are instances in which neither team scored a goal it is quite possible that shot quality is going down for the teams increasing their shot output even as the number of shots is going up.
I also thought it would be instructive to look at the different game states by breaking down the situations even further, both by home vs. road and by the amount of the deficit, i.e. one or two goals. These are the results when the teams are tied and when the home team is in the lead:
The change is significant both with a one-goal lead and with a two-goal lead. Now let's look at the results when the road team is leading:
As expected the road team also plays to the score. The difference in the results when the road team is up by two goals are particularly astounding. They outshot the home team in the first, presumbably to get a lead and were then run over territorially in the third as they protected the lead. The difference is far more pronounced than the same situation when the home team has the lead. This may be the result of the matchups. Of note, however, is that these totals represent only 13 games where the road team is up by two and only 12 games where the home team is up by two.
One final point. Certain teams compete in more of these games than others. The average number of games is 9.5. The highest totals are the Minnesota Wild with 15, the Tampa Bay Lightning with 14 and a few others at 13. The lowest totals are for the Detroit Red Wings with 3, the Columbus Blue Jackets with 6 and the Chicago Blackhawks with 7. The Oilers competed in 8.