Collins attended high school at Worcester Technical High School in Worcester, Massachusetts which compiled a record of 91–5 during his four years there. Collins was overlooked by baseball scouts because of his size, standing at only 5'7". His senior year, Collins threw a no-hitter against Auburn High School in the district championship game.

Toronto Blue Jays' general manager J. P. Ricciardi discovered Collins after he was pitching in an American Legion Baseball game. Collins began his professional career in 2007 with the Gulf Coast League Blue Jays of the Gulf Coast League Northern Division. In 7 games, Collins was involved in no decisions and had an ERA of 4.50. The next season, Collins was promoted to the Class-A Lansing Lugnuts of the Midwest League. That season, Collins went 4–2 with a 1.58 ERA, 98 strikeouts, and 14 saves in 49 games, all in relief. He was fifth in the league in ERA that season. His 14 saves ranked him third in the entire Blue Jays organization in that category. In 2009, Collins began the season with the Class-A Advanced Dunedin Blue Jays of the Florida State League. In 40 games with Dunedin, he went 7–4 with a 2.37 ERA, 99 strikeouts, and 3 saves. He was named to the Florida State League All-Star team that season. Collins was later promoted to the Double-A New Hampshire Fisher Cats of the Eastern League. Collins compiled a record of 2–3 with a 5.68 ERA, and 17 strikeouts in 9 games with New Hampshire. On the season, Collins had a combined record of 9–7 with a 2.91 ERA in 77 ¹⁄₃ innings pitched. Collins was rated as having the best curveball in the Blue Jays organization by Baseball America in 2010. Collins was selected as the Toronto Blue Jays organization's Postseason Player of the Year by MLB.com.

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