Huber was born in Cincinnati, Ohio. He attended Archbishop McNicholas High School in Cincinnati, and played for the McNicholas Rockets high school football team. He was a three-time all-conference selection and was named the Greater Catholic League's Punter of the Year following his junior and senior seasons.
Huber attended the University of Cincinnati, where he played for the Cincinnati Bearcats football team from 2005 to 2008. As a junior in 2007, and again as a senior in 2008, he was recognized as a consensus first-team All-American.
The Cincinnati Bengals selected Huber in the fifth round (142nd overall pick) of the 2009 NFL Draft, and he has played for the Bengals since the 2009 season. In four years with the Bengals through 2012, he has played 64 games and averaged 44.0 yards on 324 punts.
In week fifteen of 2013 against the Pittsburgh Steelers, Huber was blocked by Steelers linebacker Terence Garvin, which led to a fractured jaw. Afterwards, it was reported that Huber had also suffered a vertebrae fracture in his neck, and will miss the remainder of the season.