Álvarez debuted in MLB with the Blue Jays in 2011, and was traded to the Marlins after the 2012 season. He threw a no-hitter for the Marlins on the last day of the 2013 season. He was named an MLB All-Star during the 2014 season.

Álvarez signed with the Toronto Blue Jays as an international free agent, and made his professional baseball debut in 2007 with the Dominican Summer Blue Jays of the Rookie-level Dominican Summer League, pitching to a 1–2 win–loss record with a 5.61 earned run average (ERA).

Álvarez subsequently moved to the United States, where he pitched for the Gulf Coast Blue Jays of the Rookie-level Gulf Coast League, going 1–4 with a 5.63 ERA and a strikeout to walk ratio of 5.67. He was promoted to the Lansing Lugnuts of the Class A Midwest League for the 2009 season, where he was an All-Star with a 9–6 win–loss record and a 3.47 ERA. He played 2010 with the Dunedin Blue Jays of the Class A-Advanced Florida State League. He was named an All-Star and was selected to appear in the All-Star Futures Game, finishing the season with a 8–7 record with a 4.33 ERA. He started the 2011 season with Dunedin, but was promoted after two starts to the New Hampshire Fisher Cats of the Class AA Eastern League. He was again named an All-Star and made his second appearance in the Futures Game, finishing the year with a 8–4 record and a 2.86 ERA.

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