The opening cutscene, featuring the dramatic Single-Stroke Battle between Ryu's father and an unknown enemy, later revealed to be Bloody Malth, followed by Ryu reading his father's mysterious will. This, more than anything else, sets the tone for the entire series.
Ryu cradling the dying Dr. Smith in his arms. Just as he passes on, he looks up and sees a trio of men in black holding guns at him saying, "Come with us".
Again, the opening cutscene takes center stage, firmly establishing Ashtar's awesome credentials. Later events in the game would do nothing to dissuade anybody watching this scene from believing that Ashtar was one of the great villains of the NES era.
The entire cutscene at the top of the Tower of Lahja. Ryu arrives just in time to see Irene swept away by Ashtar, who then blasts Ryu with his Sword of Chaos.
After Ashtar is defeated, the cutscene switches to the Sword of Chaos vanishing into thin air. A very cool piece of foreshadowing in an era not known for such subtlety of plot.
The complete and thorough destruction of the temple at the game's climax, as it falls apart piece by piece. The visuals used to achieve this are simple, but no less impressive.
Ryu's reaction to the third boss in the final boss battle.
Ninja Gaiden Xbox
In the second game, when New York City is under attack, Ryu is seen atop a skyscraper. He pulls of his trusty bow, shoots a fiend in the throat, jumps onto it, and rides it down to street level. Don't ask why he needed to climb a skyscaper just to jump back down, it looks awesome and that's all that counts.
Ninja Gaiden II
The second fight against Genshin made the friggin Air station THEY WERE FIGHTING IN split in two, BETWEEN THEM!
In Chapter 11 shortly before fighting Genshin for the second to last time, a certain someone shows up in a cutscene, face obscured and slicing Spider Clan ninja to pieces in one quick movement, and then proceeds to hand you two rice balls. It's Muramasa.
Ninja Gaiden 3:Razor's Edge
Related to above, in the second game you fight a Helicopter on an unfinished skyscraper by shooting it with arrows until you can jump on it, slashing it up with your sword, then riding it to the ground and jumping off it before it crashes and explodes.
In the third game, Theodore appearing out of fucking nowhere, apparently not dead despite seemingly being killed by Ryu, and killing Cliff after he goes One-Winged Angel. "It's time you gave me back my daughter."