10Avatar: The Last Airbender
Avatar: The Last Airbender is a show that first comes to mind when it comes to a western show with obvious anime styles. Not only were all the fantasy nations based off of various Asian cultures, but the big mouths and eyes of the characters are directly from anime style.
Like anime, the faces were able to be very exaggerated when showing different emotions.
There were also a ton of references to Asian schools of thought such as Buddhism and Shinto. Reincarnation was also a big deal in the story, which is way more seen in anime than your average western cartoon. Obviously, The Legend of Korra is also anime inspired.