Spence enrolled at Central Arizona College, and played for their baseball team in 2008 and 2009. He was selected in the 2008 draft by the Arizona Diamondbacks and in the 2009 draft by the Los Angeles Angels of Anaheim, but did not sign either time. He transferred to Arizona State University to play for the Sun Devils for the 2010 season, where he joined Mike Leake and Seth Blair in the Sun Devils' starting rotation. He was drafted by the Padres in the ninth round of the 2010 Major League Baseball Draft, and signed with them.

He was called up by San Diego on June 21, 2011, after they designated Luis Durango for assignment to make room for him on the 40-man roster. Spence made his debut on 25 June 2011 in the top of the ninth inning against the Atlanta Braves; he faced three All-Stars in Jason Heyward, Chipper Jones, and Brian McCann, setting them down in order. He became the 29th Australian to reach the Majors and first pitcher since Rich Thompson debuted for the Angels in 2007.

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