Allen was born in Fredericton, New Brunswick, the son of Kurt and Susan Allen. Allen grew up in St. Stephen, New Brunswick, where his father worked as a high school Teacher until 1999. He started playing hockey at a young age, first as a skater, then switched to the goalie position.
On October 22, 2008, Allen was signed by the St. Louis Blues to an entry-level contract. He made his NHL debut during the playoffs on April 30, 2012, coming in to briefly relieve Brian Elliott late in the Blues' second game against the Los Angeles Kings in the Western Conference semi-finals. During the shortened 2012–13 season, Allen was recalled to the Blues and recorded his first career NHL start and win in a 4–3 overtime victory over the Detroit Red Wings on February 13, 2013.
On April 17, 2014, Allen was named the Aldege "Baz" Bastien Memorial Award winner as the American Hockey League's (AHL) Outstanding Goaltender for the 2013–14 season.
On January 6, 2017, Allen's fiancé, Shannon Adams, gave birth to their daughter, Lennon Everly Allen.
