Taseff played college football at John Carroll University and was the roommate of Don Shula. Taseff and Shula were teammates with the Browns (1951) and Colts (1953–56). Taseff later became a member of the coaching staff with Don Shula on the Dolphins (1970–1994). He has two daughters living in Baltimore, Maryland - Cindy Taseff Heath and Susan Taseff deMuth. He has two granddaughters: Celia Buchanan Heath and Eleanor Crane deMuth; and one grandson Murray R. Perkins deMuth, Jr.
He set an NFL record with the Colts in 1959 by returning a blocked field goal 99 yards for a touchdown. He finished his professional playing career in 1962 with the American Football League's Buffalo Bills.