Born in Prienzing near Cham, Bavaria in Germany, Stautner's family immigrated to East Greenbush, New York when he was three years old. He served in the United States Marine Corps before attending Boston College, where he was a four-year starter as an offensive and defensive tackle. He earned a bachelor's degree in psychology in 1950.
After being selected in the second round of the 1950 NFL Draft, Stautner played his entire career with the Steelers from 1950 to 1963. Despite being small even for his day at 6-1 and 235 pounds, he distinguished himself as one of the best defensive linemen of his era as he became the cornerstone of the Steelers bruising defense. Stautner was selected to nine Pro Bowls in his fourteen-year career and only missed six games. He also made all-NFL in 1955, 1956, 1958, 1959. He retired as the career leader in safeties with 3 and ranked third in fumble recoveries with 23. He also saw spot service at offensive guard.
In the book Passion for Sports by The Sporting News, former teammate Andy Russell shares an anecdote that highlights Stautner's elite toughness. Russell, then a rookie playing on a team that would eventually finish in fourth place in what would be Stautner's final season, sees the grizzled veteran return to the huddle holding one of his hands in the other. Russell looks down and sees that Stautner has a compound fracture of the thumb; one of his thumb bones is visibly sticking out of his skin. Russell is the only one who notices, and Stautner says only, "What's the play?" Then he plays the rest of the defensive series. When the defense returns to the sideline, Russell watches Stautner, thinking that surely he must seek medical attention now. Instead, Stautner says to someone, "Give me some tape." Then Stautner taped up his hand into a club, and he played the rest of the game.
On October 25, 1964, Stautner became the first player to have his number (70) formally retired by the Steelers.
Stautner was inducted into the Pro Football Hall of Fame on September 13, 1969, his first year of eligibility.
From 1966 to 1988, he was an assistant coach with the Dallas Cowboys, and served as the team's defensive coordinator from 1973 to 1988. He was instrumental in the development of players such as Randy White and Ed "Too Tall" Jones. Stautner stayed on with the Cowboys from 1988 to 1989 as a scout and then coached the Dallas Texans, an Arena Football League team, from 1990 to 1991. Stautner was the defensive line coach for the Denver Broncos from 1991 to 1994. While with the Broncos, he coached under both Dan Reeves and Wade Phillips.
From 1995 to 1997, he returned to Germany to become head coach of the Frankfurt Galaxy of NFL Europe. He would guide the team to two consecutive World Bowls in 1995 and 1996, winning in 1995.
Stautner was elected to the Steelers 50th anniversary team in 1982. Stautner was elected posthumously by the Pittsburgh Steeler fans to the Steelers 75th Anniversary All-Time Team in November 2007.
Stautner died at a Carbondale, Colorado nursing home at age 80 from complications of Alzheimer's disease. He is buried in Texas. He is survived by his wife, Jill Stautner of Colorado, daughters Tere Stautner of Texas and Carol Stautner Hinds, grandchildren Zachary Hinds, Alexander Hinds and Rachel Hinds of Colorado and grandchildren Danielle Stautner, Jodie Stautner and Joseph Stautner of Texas. He was predeceased by his son, Joseph Stautner.