In 2008, Allen was a redshirt freshman at Oregon State. He did not play in 2009. He then went to Louisiana Tech, and in 2010, he played in 12 games, averaging 40.8 yards on 60 punts. He had 21 punts go inside the 20-yard line. Four of his punts went over 50 yards, including a season-long punt of 66 yards.
In 2011, Allen punted 83 times and set a school record with a 46.3 yard average per punt. He ranked first in the nation with 39 punts inside the 20 yard line, and he had a season-long punt of 72 yards against Mississippi State. In the Poinsettia Bowl, he averaged 43.2 yards on five punts. Allen was named to the All-WAC first team and won the Ray Guy Award as the best college football punter.
Allen averaged 48.0 yards on 45 punts as a senior in 2012. He had 21 punts that were longer than 50 yards and 20 punts that were downed inside the 20 yard line. Allen had a career-long 85-yard punt against New Mexico State. He led the NCAA in punting yards per punt, was a unanimous All-American, and won the Ray Guy Award for the second time, the first punter to ever win back-to-back Ray Guy Awards.
Allen won his first Super Bowl with the Patriots in the 2014 season. In Super Bowl XLIX, he punted four times for 196 yards and set a Super Bowl record with a 64-yard punt.
On July 25, 2015, the Patriots signed Allen to a three-year contract extension; the NFL's Collective Bargaining Agreement allows undrafted rookies to sign extensions after two seasons. The new deal, which runs through 2018, has $6.1 million in new money, including a $2 million signing bonus and $1 million in guaranteed salary in 2016.