As a longtime NYI fan, it's easy to point to Nystrom's Cup winner in Game Six of the '80 Finals. But a contest that I will always remember fondly from that dynasty period is Game One of the '83 Finals vs. Edmonton.
This was supposed to be the Oilers year, the season in which they overtook NYI. Isles sleep-walked through the regular season, finishing second in their division. Edmonton, meanwhile, simply steamrolled through the regular season and playoffs, displaying offense like no team ever before.
Game one in Edmonton, Bossy is a scratch due to a severe cold, IIRC. From start to finish, Oilers, led by #99, Messier, Kurrri and Anderson, BARRAGE Isles, who play a bend but do not break game to PERFECTION for 60 minutes. (Messier tries to set the tone physical, with an early high speed, mid-ice cross check to the face of Isles dman Tomas Jonnson.) Somehow, the Isles find a way to get a Duane Sutter goal in transition and sit on that lead until a Ken Morrow empty netter, for a 2-0 win. Quite possibly the single best display of goaltending I've ever seen, by Billy Smith.
Set the tone for the rest of the short series, which NYI swept. It was the Isles last Cup (prior to Edmonton wrestling the Cup away the next season), and it was a masterful display of a dynasty champion in its last stages, simply finding a way to win. Needless to say, the Oilers learned a lot from that series loss as well.
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As far as other NHL games go, NYR/NJD, Game Seven in the '94 Conference Finals is also memorable.