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.
Allen played for the Midget "AAA" Canadiens under Kevin Pottle, before being drafted in the third round of the QMJHL draft to the St. John's Fog Devils. After one season with the Fog Devils, Allen was picked to play for the Under-18 World Hockey Championship in Kazan, Russia where he won gold and was named tournament MVP and top goalie. In 2008, the Fog Devils were sold and moved to Montreal, becoming the Montreal Junior Hockey Club. In 2009, Allen represented Canada at the World Junior Ice Hockey Championship in Saskatoon, winning silver after posting 4 wins and 1 subsequent loss in the final to the Americans. Allen was traded to Drummondville after the World Juniors tournament, and posted a record of 18 wins and 3 losses with a save percentage of .933% and 1.75 GAA. He was named the recipient of the Jacques Plante Trophy as goalie of the year in the QMJHL in 2009–10.
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 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 March 26, 2016, Allen shutout the Washington Capitals to help the Blues break their record for longest stretch without allowing a goal. In the 2015–16 season, he had a 2.35 GAA with a .920 save percentage.
During the 2016–17 season, Allen was pulled four times in six starts during a rough stretch of play, including a poor showing on January 19, where he allowed four goals on 10 shots against the Washington Capitals. Allen did not travel with the team to Winnipeg for the January 21 game against the Jets, and stayed home to be with his newborn daughter. He was scheduled to rejoin the team on January 23, for the remaining two games of the road trip. His play soon rebounded however, and he was named Second Star of the Week on February 13. Allen went 3–0–0 with a 1.00 goals-against average, a .967 save percentage along with his 13th career shutout against the Ottawa Senators (February 7). His two other wins were against the Toronto Maple Leafs (2–1 OT, 31 saves, February 9), and Montreal Canadiens (February 11).
In 2016, Allen created a painting of the St. Louis skyline for the Kidney Foundation of Canada Atlantic Branch's "A Brush of Hope" Celebrity Art Auction. His painting was auctioned off on ebay.ca in a ten-day online auction.
On January 6, 2017, Allen's fiancé, Shannon Adams, gave birth to their daughter, Lennon Everly Allen.