Albers went to high school at John Paul II Collegiate. He was selected by the Milwaukee Brewers in the 12th round (346th overall) of the 2004 Major League Baseball draft, but did not sign. Albers enrolled at the University of Kentucky, where he played college baseball for the Kentucky Wildcats baseball team in the Southeastern Conference of the NCAA Division I. At Kentucky, Albers pitched in 81 games (fifth most in Wildcats history), recording 20 wins (fourth most in Wildcats history), and 12 saves (third most in Wildcats history).
The San Diego Padres selected Albers in the 10th round (315th overall) of the 2008 Major League Baseball draft. After signing with the Padres, Albers made his professional debut with the Arizona League Padres of the Rookie-level Arizona League, making five appearances. Albers missed the 2009 season after tearing an ulnar collateral ligament in his pitching elbow during spring training, which required Tommy John surgery. The Padres released Albers, and he pitched for the Québec Capitales of the independent Can-Am Association in 2010. With Québec, Albers had a 3-0 win–loss record, 17 saves and a 1.40 earned run average (ERA). Albers tried out for teams in Arizona during spring training in 2011, and drove to Florida to try out for the Minnesota Twins. The Twins signed Albers to a minor league contract, and he pitched for the Fort Myers Miracle of the Class-A Advanced Florida State League and New Britain Rock Cats of the Double-A Eastern League in 2011.
Albers played for the Canadian national baseball team. In 2011, he participated in the 2011 Baseball World Cup, winning the bronze medal, and the Pan American Games, winning the gold medal. Albers pitched 6 2⁄3 innings in the gold medal game. Along with his teammates, Albers was inducted into the Canadian Baseball Hall of Fame in 2012. Albers was also named Saskatchewan's male athlete of the year.
Albers was called up by the Twins on August 3, 2013, to replace fellow Canadian Scott Diamond, who was optioned to Triple-A. At the time of his call-up, Albers had posted a record of 11–5 and a 2.86 ERA with the Rochester Red Wings, with an International League-leading 116 strikeouts.
On August 6, 2013, Albers made his Major League debut and earned the win, pitching 8 1⁄3 scoreless innings against the Kansas City Royals and yielding just 4 hits and 1 walk while striking out 2. In his second career start on August 12, Albers threw his first complete game and shutout, defeating the Cleveland Indians 3–0. Albers gave up 2 hits, no walks, and struck out 2 batters. In 2013, Albers posted a 6–13 record with a 5.89 ERA.
On January 30, 2014, the Twins released Albers, allowing him to sign with the Hanwha Eagles of the Korea Baseball Organization.
The Eagles declined Albers' option for 2015, making him a free agent.
On December 16, 2014, Albers signed a minor-league contract with the Toronto Blue Jays that included an invitation to spring training. He did not make the team, and was assigned to the Buffalo Bisons, the Jays' Triple-A affiliate. Albers was the Opening Day starter for Buffalo. Albers had his contract purchased by the Blue Jays on May 1, after Daniel Norris was optioned to Buffalo and Maicer Izturis was moved to the 60-day disabled list. After pitching in one game, he was optioned back to Buffalo on May 2. Albers elected free agency on November 6.
On March 3, 2016, Albers signed with the Lancaster Barnstormers of the Atlantic League of Professional Baseball.
On April 27, 2016, Albers signed a minor league deal with the Minnesota Twins. On August 11, Albers was brought up to the Twins from the Triple-A Rochester Red Wings. Pitching against the Houston Astros the same day, he made the longest relief appearance of any major league reliever in 2016 by both number of innings pitched (6) and pitches thrown (108). He was optioned back on August 14, and recalled on August 26. He was assigned outright to Rochester on October 17. He elected free agency on October 18.
In December 2016, Albers signed a minor league contract with the Atlanta Braves. He went 12–3 with a 2.61 ERA and 115 strikeouts for the Triple-A Gwinnett Braves of the International League.
On August 11, 2017, Albers was traded to the Seattle Mariners in exchange for cash considerations.