Hannah was raised in Albertville, Alabama, and excelled at football, wrestling and track at Baylor School in Chattanooga, Tennessee. He won an individual national championship in wrestling at the National Prep Championship in 1967. His high school coach in football, wrestling and track was Luke Worsham, whom Hannah credited in his induction speech at the Pro Football Hall of Fame: "I want to talk to you about Major Luke Worsham who was my high school coach. He is the guy who really taught me and showed me what love was all about. Because he would shoulder my problems, he would always stand by me to offer me encouragement when things got tough and when things were going awry he would make sure he would correct me and get me right back on track." Hannah played his senior season of high school football at Albertville High School, where he graduated in 1969.
Hannah played tackle and guard for the University of Alabama under Coach Paul "Bear" Bryant from 1970 until 1972, and earned All American honors twice, in 1971 and then was a Consensus All American his senior year in 1972. During his tenure Hannah was part of an SEC championship-winning team. He was named to the University of Alabama All-Century Team and also to the Alabama 1970s All-Decade team.
During his time at Alabama he also participated in wrestling, the shot put, and the discus throw. Hannah was inducted into the College Football Hall of Fame in 1999. Bryant would later say that Hannah was the greatest lineman he ever coached.
Hannah joined the Patriots in 1973 as the 4th overall pick in the 1973 NFL Draft. He played his entire professional career in New England. While considered somewhat short by NFL standards, Hannah made up for this with great speed and quickness as well as powerful legs. Hannah excelled as a pass protector, run blocker and as the pulling guard on sweeps. Hannah's commitment level to football was very high and he expected the same from each of his teammates, sometimes becoming quite angry if he did not feel that they were complying. Hannah started the first thirteen games of his rookie season of 1973 until a freak leg injury forced him to miss the final game of the year. Along with tackle Leon Gray, the two formed what was generally considered the best guard/tackle tandem in the NFL during the mid to late 1970s. Gray and Hannah also combined with tight end Russ Francis to form one of the strongest left-side trios in the league. Hannah anchored the 1978 offensive line that set a still-standing NFL record with 3,165 rushing yards. Hannah missed only five games out of a possible 191 because of injuries during his career. He also missed the first three games of the 1977 season due to he and Gray both holding out because of contract disputes. The Patriots finished with a winning record seven times and had only three losing seasons during Hannah's thirteen-year career. In 1985, Hannah helped guide the team to its first AFC title and Super Bowl appearance. Hannah retired from the NFL after playing in Super Bowl XX.
Hannah was named All-Pro 10 times (1976–1985) and All-AFC 11 times (1974, 1976–1985). He was also selected to play in 9 Pro Bowls. He was voted the Seagram's Seven Crowns of Sports Offensive Lineman of the Year Award in both 1978 and 1980 and was the winner of the NFLPA - Coca-Cola Offensive Lineman of the Year Award (selected by a vote of NFL players) for four straight years (1978–81). He is also one of the few players to have been named to the NFL All-Decade Team twice, as Hannah was selected to both the 1970s and 1980s All-Decade Teams (joining an elite group of players who have achieved this including Walter Payton). Hannah was also selected to the NFL 75th Anniversary All-Time Team, being the #1 guard in the team.
In 1991, he became the first Patriots player to be inducted into the Pro Football Hall of Fame. He and Andre Tippett are the only members of the Hall of Fame to have spent their entire career with the Patriots. In 1999, he was ranked number 20 on The Sporting News' list of the 100 Greatest Football Players, the highest-ranking Patriot, the highest-ranking guard, and the second-ranked offensive lineman behind Anthony Muñoz.
Hannah became an assistant coach at Governor Dummer Academy in Massachusetts, leaving to accept the head coaching job at Somerville, Massachusetts, High School in 2004. While concurrently serving as the city's youth development coordinator, Hannah led the Somerville team through one winless season. He left to become a special assistant coach at his alma mater, Baylor School in Chattanooga, Tennessee, in 2005. He resigned from his coaching position at the conclusion of Baylor's 2005 football season.