Vanney, who resigned from his position at Toronto in December, takes over after a difficult 2020 season for Galaxy.
The former Galaxy defender said he was looking forward to the opportunity as he replaces Guillermo Barros Schelotto.
"I am extremely excited about the opportunity to coach the LA Galaxy," Vanney said.
"The LA Galaxy are an iconic club with a rich history in Los Angeles, MLS and North American soccer over the last 25 years.
"I am committed to working tirelessly for this club, our fans and Los Angeles to win a championship. I look forward to this new challenge in my career and am eager to get started."
Galaxy have missed the playoffs in three of the previous four seasons, winning just six games in 2020.
Vanney led Toronto to a treble in 2017, winning the MLS Cup, Supporters' Shield and Canadian Championship.
"Greg is a part of the history of the LA Galaxy and we are confident that he is the right person to lead our team moving forward," Galaxy general manager Dennis te Kloese said.
"He has deep knowledge of this club and Major League Soccer as a player and a coach and has proven to be a consistent winner with a championship pedigree. His accolades as Toronto FC's manager are evidence of his ability to create and sustain success in our league.
"We look forward to working with Greg to continue to build a team that will compete for trophies."