Bell qualified as one of the two fastest third-place finishers to secure her place in Tuesday's final along with compatriots Jemma Reekie and Keely Hodgkinson.
Reekie finished second in her semi-final behind Jamaica's gold medal contender Natoya Goule while Bell produced a great run to finish third behind Athing Mu of America and Ethiopia's Habitam Alemu in her heat.
Hodgkinson won European indoor gold in March and produced a stunning finish to move from sixth into first in the third semi-final to make it a hat-trick of British athletes in next week's final.
Bell told the BBC in the wake of her qualification: "It is a dream come true, I did not expect to be on this journey and now I am on this journey and I said regardless of outcome I wanted to enjoy every step. At least I wanted to walk away with a PB or a qualifying time for next year.
"Never in my wildest dreams did I think I would be an Olympic finalist."
Hodgkinson added: "I am so happy to qualify, to have three I don't think that has ever happened before and all of the girls deserve it. I am going to go out and do the best I can, if that brings out a medal then so be it."
Bell was praised by Sheffield's 2012 Olympic champion Jessica Ennis-Hill, who said: "Alex Bell is in a position of opportunity and she has taken it with both hands running the time she has.
"Jemma Reekie is in the form of her life, she has got the tactics right and they are all in a fabulous position going forward."
Meanwhile, Dina Asher-Smith’s hopes of individual Olympic glory are over after heartbreak in the 100 metres semi-final prompted her to withdraw from the 200 metres competition.
The Londoner had been tipped to become the first British woman to take an individual sprint medal since Dorothy Hyman in 1960.
But she could only manage third in her semi-final to crash out in Tokyo, before revealing the hamstring injury she suffered last month had badly hampered her preparations.