Ms Montague, who had a 33-year career in the Navy, developed a computer program that created rough drafts of ship specifications.
She told the Arkansas Democrat-Gazette in 2012 that it allowed the Navy to cut the time it took to build a ship's draft design from two years to 18 hours and 26 minutes.
She died on Wednesday at a hospital in Little Rock, the newspaper reported. Her son, David Montague, said she died of congestive heart failure.
Raye Montague, a civilian engineer, fought discrimination during her career. When a co-worker would not show her how to operate a computer, she taught herself.
"I had to run circles around people, but when they found out I really knew what I was talking about they came to respect me," Ms Montague told the newspaper. "I worked long hours and travelled for the job because I couldn't say I wanted the same wages as the guys if I couldn't. I had to do all the same things, within reason, that they did."
Her son said: "She always made it a point to just try to meet every challenge with a smile."
Ms Montague's work became widely recognised after the release of the movie Hidden Figures in 2016. The film told the story of the work done by black female mathematicians during the early days of Nasa.
A 2017 article on the Navy's website about Ms Montague noted she was known as the Navy's "hidden figure".
Ms Montague was born in 1935 in Little Rock and grew up in segregated Arkansas. David Montague said his mother was unable to pursue engineering in college because the University of Arkansas in Fayetteville would not accept black students at the time.
So Ms Montague attended Arkansas Agricultural, Mechanical and Normal College, now the University of Arkansas at Pine Bluff, where she studied business instead.
After graduating in 1956, she went to Washington DC, determined to find a job related to engineering. She started as a clerk typist with the Navy and worked her way up to become its first female program manager of ships.
"She was involved with STEM (science, technology, engineering and math) before it was sexy," said Anna Beth Gorman, executive director of Women's Foundation Arkansas.
Ms Montague, who was awarded the Meritorious Civilian Service Award by the Navy, moved home in 2006 to be closer to her son David and his family. She was inducted into the Arkansas Women's Hall of Fame this year.
David Montague said that despite her busy work schedule and career goals, his mother, who was divorced, always made sure to take him to Boy Scout meetings and rarely missed a bowling match.
"I think that's what she'd want people to know," he said. "That she was a real person."
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