Myth: With Grant Fuhr making the big saves in crucial situations, the 1980s Edmonton Oilers won a lot of 5-4 and 6-5 games.
Truth: The Oilers rarely played in 5-4 or 6-5 games, and even when they did they were just as likely to lose as to win.
In the popular memory, Grant Fuhr was a goalie that bore down in the final moments of the game, shutting the door on the opposition to preserve 6-5 shootout wins for the Oilers. Unfortunately, this perception doesn't match the historical record. From 1982 to 1989, the Edmonton Oilers won a playoff game 5-4 a total of only four times, and two times they ended up on the wrong end of the 5-4 score. They won 6-5 just twice while losing three times by the same result.
Edmonton did have a very good playoff record in one goal games (26-9). That was partly because they were very good in overtime, going 9-3. It is easy to attribute a strong overtime record to the goalie, but these are Gretzky's Oilers we are talking about, and they tended to score goals quickly. Five of the 12 overtime games were decided before the teams even played 65 seconds of OT. Only four of them went longer than the 4 minute mark. Despite the impressive record, Edmonton's goalies (mostly Fuhr) had a mediocre 4.00 GAA in overtime, nearly a goal higher than their GAA in the first 60 minutes. Edmonton did well in one-goal games, but only about a quarter of their games were decided by one goal because they so often blew out their opponents. Edmonton also did very well in games decided by two goals or more (58-21).
I looked at Grant Fuhr's regular season results from 1985-86 to 1988-89 on hockeygoalies.org to try to see if they matched the playoff numbers. In games decided by 5-4 scores, the Oilers had 6 wins and 5 losses. In 6-5 games, the Oilers won twice and lost four times. I also checked out the results for 6-4 and 7-5 games just to see if I was missing something, but in those games combined the Oilers had 7 wins and 5 losses. The Oilers again had a good one-goal game record (34-21), but it was actually worse (.618) than their record in games decided by 2 goals or more (.640).
Out of Fuhr's 483 regular season and playoff games in the 1980s with Edmonton, he won only 14 of them by 5-4 or 6-5 scores, and 11 of those wins came in overtime, where Edmonton won a lot of games because their offence was so quick to score. Fuhr also lost 13 games by one of those two scores, mostly in regulation time, meaning that Edmonton wasn't really any more likely to win the close high-scoring shootouts than their opponents were.
There is some evidence that the Oilers won more games than they should have, both in the playoffs and in the regular season. From the Pythagorean expected win formula (based on goals scored and goals allowed), the Oilers would have been expected to have won 68.7% of their playoff games and 63.4% of their regular season games between 1982 and 1989. They actually won 73.7% of their playoff games and 66.6% of their regular season games. However, Edmonton platooned their goaltenders, using both Fuhr and Andy Moog. The two goalies had similar stats: Moog's GAA with Edmonton was 3.61, Fuhr's was 3.76 over the same time period. Moog also had a better winning percentage than Fuhr did with the same teams, in both the regular season and the playoffs.
So even if Fuhr was a clutch goalie, then so was Andy Moog, since Moog had better stats and a better winning percentage. What is more likely, however, that one team had two excellent, clutch goalies that led them to repeated Cups, or that the high-scoring Oilers dynasty made their goalies look like winners? Based on the evidence, the second option looks like a pretty safe bet.