Elden Francis Curtiss was born on June 16, 1932, in Baker City, Oregon, as the eldest of four sons to Elden F. and Mary A. Neiger Curtiss. He received his early education through high school locally before studying at St. Edward Seminary in Kenmore, Washington. 

Ordained a priest for the Diocese of Baker on May 24, 1958, by Bishop Francis Peter Leipzig, he served in parishes across Lakeview, La Grande, and Jordan Valley, and as a hospital chaplain, while pursuing advanced degrees including a Master of Divinity and a Master of Arts in education administration from Fordham University, the University of Portland, and the University of Notre Dame.

Elden Francis Curtiss advanced in diocesan roles as director of information, superintendent of schools, and faculty member at Mount Angel Seminary, later serving as its president-rector from 1972. 

On March 4, 1976, Pope Paul VI appointed him the seventh Bishop of Helena, Montana, with his episcopal consecration occurring on April 28 by Archbishop Cornelius Michael Power, and he adopted the motto "That We May All Be One" from John 17:21. During his 17-year tenure there, he acted as chancellor of Carroll College and contributed to various national Catholic committees.

Pope John Paul II named Curtiss Archbishop of Omaha, Nebraska, on May 4, 1993, succeeding Daniel E. Sheehan, with installation on June 25.

He led a successful capital campaign funding priest retirement homes, St. Cecilia Cathedral repairs, Catholic high school tuition subsidies, and seminary endowments, while serving on USCCB committees for marriage, family life, and vocations, as well as boards for the Catholic Church Extension Society and Kenrick-Glennon Seminary.

His leadership faced scrutiny over handling certain clergy abuse cases, including reassignments that drew later criticism.

Elden Francis Curtiss submitted his resignation upon turning 75 in 2007, which Pope Benedict XVI accepted on June 3, 2009, appointing George J. Lucas as successor. In retirement, he reflected on the Church's improved response to sexual abuse crises, marking 65 years as a priest in 2023 at age 90. He remains archbishop emeritus of Omaha.