Lyon, home to young Matildas star Ellie Carpenter, split with coach Kean-Luc Vasseur after their recent quarter-final elimination from the UEFA Women's Champions League at the hands of PSG.
Lyon have won the last five editions of the Champions League consecutively and have a record seven total titles.
New manager Bompastor is a former Lyon and France player who spent six seasons at the club as a midfielder and defender.
For the past eight years she's been director of Lyon's women's academy.
She has signed a two-year contract with the French juggernauts.
"I am a competitor, and I like challenges," Bompastor said.
"We have everything we need to win the title and rediscover the [Lyon] DNA on the pitch.
"I'm going to put that into place."
Last season, his first in charge, the departing Vasseur was named UEFA women's coach of the year.
Lyon are currently one point behind rivals PSG in the French domestic league, in danger of losing that crown too.
