He played college football for the Oregon State Beavers and the Louisiana Tech Bulldogs.

[2] In 2008, Allen walked on at Oregon State along with fellow punter Johnny Hekker.

Allen was named to the All-WAC first team and won the Ray Guy Award as the best college football punter.

[11] He signed a standard three-year contract as an undrafted free agent with the New England Patriots on April 27, 2013.

During training camp, Allen beat out veteran Zoltán Meskó, who was released at the final roster cutdown.

[17][18] His performance earned him AFC Special Teams Player of the Week honors.

[19] On February 5, 2017, Allen was part of the Patriots team that won Super Bowl LI.

In the game, Allen punted four times for 166 yards as the Patriots defeated the Atlanta Falcons by a score of 34–28 in overtime.

[21] The next year, his former college teammate Johnny Hekker broke his Super Bowl record for the Los Angeles Rams, though the Patriots would win that game and give Allen his third Super Bowl championship.

[24] On November 4, 2019, Allen signed with the Atlanta Falcons, making him the team's fifth Punter of the season.