Margaret Elliott was born in 1906 in Daming, Hubei Province, China, where her parents, Rev. Walter Scott and Dr. Eleanor Edwards Elliott, were missionaries. Elliott was the third eldest of six children (Curtis, Nathan, Margaret, Eleanor ("Ruth"), Frances, and Edwards). She spent her early childhood in Changsha, Hunan Province, from where her father coordinated the distribution of Bibles and Christian literature through colporteurs of the American Bible Society. The Elliott children often accompanied their father on his tours of the country, gaining early experience with the Chinese language and culture. When old enough, Elliott joined her older brothers at the Kuling American School in Jiangsu Province, where she received her early formal education.

When Elliott was about ten, the Elliotts returned to the United States for their furlough in Claremont, California. While the Bible Society sent the Elliott parents to work in France in the Chinese community there, their children remained in California with an aunt to continue their education. From 1924 to 1926, Elliott attended Biola College. In 1926 she transferred to Wheaton College, from which she graduated in 1929, majoring in education. Also in 1929, she applied to and was accepted by China Inland Mission (CIM). She had a longer-than-usual stay at CIM's language school due to illness and the shortage of senior missionaries to supervise new workers. In 1931 she was sent to her first station at Shucheng, Anhui Province. Throughout her missionary career, Elliott focused her work on teaching the Bible, primarily among Chinese women and children, with the goal of establishing healthy Chinese congregations which could lead themselves. In 1933, having achieved senior missionary status, she was transferred to Tungcheng, Anhui; she was senior missionary at the station, supervising her sister, Eleanor "Ruth" Elliott, who had also come to China with CIM. In 1934, Elliott was returned to Shucheng.

In 1935, she married Vincent Crossett. Together, the Crossetts went to Zhengyangguang, Anhui, also working in nearby Huoqiu and the surrounding countryside. The Crossetts were then transferred to Hoqiu to continue their work in the area, which was hindered by the presence of Japanese troops. In 1939, the Crossetts went to the United States for their first furlough. Returning to China in 1941, they were again stationed at Zhengyangguang. As World War II continued and Japanese troops threatened Anhui Province, Mrs. Crossett and daughters left Mr. Crossett behind in 1944 to travel west and south to a safer station. She was temporarily posted at Fengxian, Sichuan Province. She was reunited with Mr. Crossett in 1946 and they were then stationed at Lu Xian, Sichuan. In 1947 the Crossetts took their second furlough, living in Wheaton, Illinois, while Mr. Crossett worked on a masters degree in the Wheaton College Graduate School; Mrs. Crossett took courses in the Christian Education Department.

Due to civil war conditions in China and the needs of the their daughters, the Crossetts resigned from China Inland Mission in 1948 and accepted a pastorate in Honolulu. In 1961, having seen the completion of their daughters' college educations, they reapplied to the mission, by then renamed Overseas Missionary Fellowship (OMF). They were sent to Taiwan and stationed at OMF's mission home in Tainan. For two years, the Crossetts were responsible for administrative tasks and hospitality. In 1963 they were assigned to teach at the Yu Shan Theological Institute, a school for a Taiwanese tribal people located in Hualien. With a furlough in 1965, the Crossetts continued teaching at the Institute until their 1970 furlough. Upon returning to Taiwan in 1971, Rev. Crossett worked on preparing audio taped instructional materials for use by tribal groups in Taiwan. In 1973, the Crossetts retired from the mission due to Mrs. Crossett's heart attack.

The Crossetts continued their Bible teaching ministry in the United States as Mr. Crossett pastored a small community church in Alto Pass, Illinois. They later moved to Wyanet, Illinois, from where Mr. Crossett worked as an area evangelist for the Reformed Presbyterian Evangelical Synod and Orthodox Presbyterian Church, and served as a volunteer regional representative for OMF. In 1986, the Crossetts moved to OMF's retirement home in Lancaster, Pennsylvania, where they continued to reside in 1988.

The Crossetts had two daughters, Virginia and Margaret, born in 1937 and 1939 respectively.