Koteas was born in Montreal, Quebec, Canada to a father who worked as a mechanic for the Canadian National Railways and a milliner (hatmaker) mother. His parents are both of Greek descent. He can speak Greek and French fluently. Koteas attended Vanier College in Montreal, Canada before leaving to attend the American Academy of Dramatic Arts in New York City in 1981, of which he is a 1983 graduate. He was a member of the Academy's 1983–1984 Production Company. He also attended the Actors' Studio in New York City, where he studied acting under Ellen Burstynand Peter Masterson. During his early years, Koteas washed dishes and bussed tables in New York for $1 an hour. At the time, he was not a U.S. citizen and therefore was not allowed to work in that country.

While at the AADA, Koteas played Father Rangier in the school's production of "The Devils" adapted by John Whiting from the Aldous Huxley novel. He was also Paris in "The Golden Apple" a musical by John Latouche and Jerome Moross.

Koteas is known for his film portrayal of Casey Jones in the popular Teenage Mutant Ninja Turtles movies and for the character Thomas Daggett in the American film The Prophecy, which also starred Christopher Walken, Viggo Mortensen, Eric Stoltz (with whom Koteas also appeared in Some Kind of Wonderful) and Virginia Madsen(with whom Koteas also appeared in The Haunting in Connecticut).

Koteas had a role in the Denzel Washington thriller Fallen as the demon-possessed serial killer Edgar Reese. He appeared in John Hughes' Some Kind of Wonderful, Atom Egoyan's Exotica, Ararat and The Adjuster, Terrence Malick's The Thin Red Line and David Cronenberg's Crash. Koteas also made an appearance in Season 4 of The Sopranos as Dominic Palladino, and in the Season 2 finale of House, in which Koteas plays a man who shoots Dr. Gregory House. Elias has also appeared in The Greatest Game Ever Played, a Disney biography about a young golfer, as well as the thrillers Skinwalkers in 2006, Zodiac and Shooter in 2007. In May/September 2008 he played the role of "Joe," a bank robber, in the season 4 finale (and season 5 premiere) of CSI: NY.