The 29-year-old joins the club on an 18-month deal after a successful spell on trial at Rangers in November.
This will make Bala the first Indian woman to become a professional footballer anywhere in the world and she also becomes the Rangers' first Asian international footballer.
Bala is the current top scorer for the Indian national women's team, netting an impressive 52 times in 58 games since 2010, also making her the top international goal scorer in the South Asian region.
She has also served as her national team captain in a distinguished international career, which began when she was called up aged just 15.
Bala also has a prolific goalscoring record at the domestic level with over 100 goals in 120 games. She has been the top scorer in the Indian Women's League for the past two seasons and has also been named as All India Football Federation (AIFF) Women's Player of the Year twice, in 2015 and 2016.
Speaking of her delight at signing for Rangers, Bala said: "To play my football in Europe with one of the biggest clubs in the world is something I could never have dreamed of. "I hope my move to Rangers serves as an example to all the women footballers back home in India who dream of taking up the sport professionally.
"I'm looking forward to making the most of the top-class facilities and coaching and I am certain I will benefit immensely from the standard of training and competition. I'm very grateful to Amy McDonald, the coaching staff and the entire management at Rangers for believing in me. Also, this move wouldn't have been possible without Bengaluru FC who have been instrumental throughout. I can't wait to get to Glasgow, join my teammates and give this my best shot."
Amy McDonald, Women's and Girls' Football Manager at Rangers FC said: "We are thrilled to welcome Bala to Rangers. She is an exciting signing on so many levels. Bala is a playmaker who likes to play as a number 10 and we believe she will contribute goals and assists to the team. She will provide us with an attacking threat and a versatility that we can utilise to our benefit going into the 2020 season. Bala is already a role model for girls across India and they will now be able to see her travel across the world to become a professional footballer. Her move can be inspirational for players everywhere to show them where football can take them and what it can help them achieve."
( With inputs from IANS )