Wistert, who was born in Chicago, Illinois, was from a Lithuanian family. His father, Kazimer Wistert, was a Spanish-American War veteran who was later killed in the line of duty while working for the Chicago Police Department. The story of the Wistert brothers at Michigan began when Whitey's Carl Schurz High School classmate John Kowalik was invited to visit the University of Michigan. At the time, Whitey Wistert had a factory job where he was involved with building Majestic radios. Kowalik took Whitey with him on his visit to Ann Arbor, and according to Alvin, "that's how it started: the Wisterts of Michigan."
And if I'm not mistaken I think this is unprecedented in the annals of college football: that three brothers all would go to the same school, all played football. All played tackle, all wore the same number 11, all made All-American. Two of us played on four national championship teams. And all were inducted into the College Football Hall of Fame. – Alvin Wistert
After graduating from Foreman High School, Wistert became the second of the Wistert brothers to play for Michigan where he wore number 11 like his brothers and played from 1940 to 1942. He was a consensus All-American and team MVP in 1942. He played in the 1943 East–West Shrine Game. He is well remembered, among other things, for his exploits in a 1942 game against the Notre Dame Fighting Irish football team, and he was inducted into the College Football Hall of Fame in 1968, one year after his brother Francis. In 1981, he was named to the University of Michigan Hall of Honor in the fourth class of inductees alongside his brothers. Only five Michigan football players earned this honor before him.
Michigan posted a 20–5–1 record during Wistert's three years on the team. In 1940, the team's only loss in its eight-game season was to the eventual national champion Minnesota Golden Gophers football. The Wolverines followed that season with 6–1–1 and 7–3 marks in the next two years. Wistert served as captain of the College All-star team that beat the Sammy Baugh-led National Football League champion Washington Redskins, 27–7, in Chicago. He was the only one of the three brothers not to play on a national championship squad at Michigan.
After being drafted in the fifth round by the Philadelphia Eagles and signing for $3800 ($51,790 today), he encountered animosity from veteran players for having signed such a large contract (though they thought he signed for $4500). He earned All-Pro honors in eight (five by consensus) of his nine seasons. As a two-way player, he played his entire nine-year career for the Philadelphia Eagles (who operated as a merged team with the Pittsburgh Steelers for one season during World War II.) The National Football League had no All-Star games between 1943 and 1950. Thus, although Wistert was a perennial All-Pro selection, it is difficult to compare him to more modern players who are often measured by Pro Bowl invitations. He served as Eagles captain for five consecutive seasons, from 1946 to 1950, and was named All-Pro in each season. In Wistert's next to last season he was selected to the first Pro Bowl. In his final season, he recovered three fumbles. The Eagles won the 1948 and 1949 National Football League Championships with Wistert. These were the only consecutive National Football League champions to win by shutout. His number 70 was retired by the Eagles in 1952. According to his College football hall of fame biography, during his career he started every game the Eagles played except for the 1950 season opener against the Cleveland Browns. However, according to other statistical databases he only started about 2/3 of his career games. His is one of only seven retired Eagles Jerseys. Wistert lamented not having been enshrined in the Pro Football Hall of Fame and on Philadelphia Eagles Honor Roll:
The two things that would really make my career complete is to be inducted into both the Pro Football Hall of Fame and the Philadelphia Eagles Honor Roll.
Wistert was inducted into the Philadelphia Eagles Honor Roll on September 29, 2009 along with Randall Cunningham.
After football Wistert became a successful life insurance salesman. He sold insurance for 40 years. Among the places he has lived since retirement are California and Grants Pass, Oregon. He was married to his late wife Ellie for 61 years and has three daughters (Pam, Dianna and Kathy) and three grandchildren. At one point he was both coaching football at Riverside High School in Riverside Township, New Jersey and playing professionally. According to brother Alvin, their father "was born Casmir Vistertus and he Anglicized it when he came to America to Wistert."