Scheuer will take over after former England coach Hope Powell stepped down at the end of October after five years in charge.
During his three seasons at Bayern, the 44-year-old German coach helped the team win the Women's Bundesliga title and guided them to the semi-finals of the Women's Champions League in 2021.
Scheuer will be tasked with rejuvenating the fortunes of Brighton, who are second-bottom in the 12-team WSL at the winter break.
"I had good talks with (technical director) David Weir. The first time we spoke it was clear for me that I wanted to come here and build a team which can grow and move up the table," said Scheuer, who has agreed a deal until the summer of 2026.