The fee broke the previous record of £685,000, when Bay FC signed Racheal Kundananji last February.
Chelsea set the British transfer record in signing Mayra Ramirez for £384,000 in January 2024.
Girma told BBC Sport: “I’m really excited to play alongside Millie (Bright).
“There are so many players in this team that have so much experience, a lot of young players who have been amazing as well, so [I am] just really excited to be in with the group.”
Defender Bright told BBC Sport: “We’re really lucky to have her. We feel really grateful that she chose us.
“It sends out a statement to the rest of the league and Europe about the direction we are going in and the titles that we want to win.
“She is going to be a fantastic addition and we can’t wait to get started with her.”
Chelsea manager Sonia Bompastor said after the Arsenal game that Girma has a “small injury” and would not be fit for the trip to Aston Villa on 2 February.
Girma helped San Diego win the National Women’s Soccer League (NWSL) Shield in 2023 and was named US Soccer’s female player of the year.
She was part of the USA side that won gold at the 2024 Olympics in Paris.
Girma is Chelsea’s first signing of the January transfer window after they fought off competition from Lyon.
Chelsea made her a priority target after losing defender Kadeisha Buchanan to a season-ending knee injury.