Baxter was raised in Sydney, Australia and commenced his career as an editor there. He had been noted primarily for his work editing commercials, and co-founded a commercial editing firm named Final Cut. In 2004, Baxter joined Angus Wall's firm Rock Paper Scissors, which was also responsible for commercial work. Baxter then worked with Wall as an "additional editor" on Fincher's film Zodiac. When Wall was chosen to edit The Curious Case of Benjamin Button, he proposed to Fincher that Baxter be the co-editor, and they have since worked together on Fincher's subsequent films.
In 2014, Baxter's next edited film was Gone Girl, also directed by David Fincher, and was his first film to not be co-edited by Angus Wall. Coincidentally, the film is also the first for which he was not nominated for the Academy Award for Best Editing.