The Tale of Zabuza Momochi and Haku is full of them. It opens with an animated cutscene of Zabuza and Kakashi proving why they're easily among the best of their respective villages. Zabuza then proceeds to give Kakashi a Hannibal Lecture that seemingly shakes him slightly. Kakashi, as opposed to the 3rd Hokage, then narrates the tale, the awesome coming in how he brings up the internal conflict with in Zabuza. The story ends with Zabuza taking out nearly all of Gato's men. Upon killing Gato the rest of the thugs run away in fear (who could blame them). Additional awesome comes as the story is even more of a Crowning Moment of Heartwarming/Tearjerker than the original as Zabuza has flashbacks of his and Haku's time together which push him on during his last stand against the thugs.
The Tale of Gaara's opening definitely counts as in this version Gaara and Sasuke actually fight, and it is AWESOME.
No matter how short it is, this troper loves the opening cutscene battles, as well as battles like the one against Pain/Pein
The Tale of Madara Uchiha starts with an episode-quality animation of Madara and Hashirama's battle in the Valley of the End. And it is awesome.
Alot of the ultimate jutsus are freaking cool and can be seen here. Two notable examples that come to mind are Kakashi's twin lightning shiver (1:35), and snake!Kabuto's impure world resurrection (6:43).